On the analysis of movement smoothness

Movement intermittency can be observed in upper-extremity movements of infants below 6 months [3], in slow movements [20, 31], movements requiring accuracy [32], and in patients with neurological conditions such as stroke [5], multiple sclerosis [33], Parkinson’s disease [34, 35] etc. Reaching arm movements of healthy adult subjects have almost straight line trajectories with bell-shaped speed profiles [11]. However, the same task performed by stroke subjects can be highly intermittent, with the hand accelerating and decelerating several times on its way to the target [5] as shown in the simulated example of Additional file 1: Figure S1.

An important point to note here is that intermittencies can occur in a movement due to two distinct set of factors:

Given these two distinct factors that can contribute to movement intermittency, smoothness could be used as a criterion for differentiating between (a) control abilities of different subjects when the task is controlled; and (b) different types of tasks when skill is controlled. The former type of analysis is what one typically encounters in motor learning and neurorehabilitation literature. The latter analysis was discussed by Hogan and Sternard in [28] where they propose mean squared jerk as a criterion for differentiating between discrete and rhythmic movements. However, this can be done only if the two movements were performed by the same subject or by two subjects that are equally skilled.

A smoothness measure λ _s is a function

that characterises a movement’s smoothness, where M _m represents some measured information about the given movement, e.g. kinematics; and T represents the task, e.g. point-to-point reaching, target tracking, or some spatio-temporal constraints. The task-dependence of smoothness is indicated by T in Eq. 1.

A good estimate of movement smoothness requires a measure that is valid, sensitive, reliable and practical. Validity is the most important property, which refers to whether or not a given measure estimates the factor it is intended to measure [38]. Sensitivity and reliability deal with how well a given measure is able to quantify a movement related factor. Sensitivity is essential for a measure to resolve real but small differences in a factor. A measure must have good sensitivity, particularly in the physiological range that covers the entire spectrum of healthy and un-healthy movements. Reliability is directly related to the robustness of the measure to measurement noise¹, i.e. the amount of variability introduced in the measure by a given level of measurement noise. Practicality refers to issues related to the practical implementation of the measure, which needs to consider factors such as computational complexity of the measure (e.g. how does the computational time increase with increase in data size).

For movement smoothness, a valid measure must be independent of the movement amplitude and duration, i.e. dimensionless. Smoothness is measured through the amount of movement intermittency, which is directly related to the movement’s temporal organization or coordination. Thus, a valid smoothness measure must change monotonically to changes in movement intermittency. For example, if a given movement can be thought of as a superposition of discrete submovements, then an increase in the number of submovements or the interval between two successive submovements must result in decreased movement smoothness. This can be understood from Additional file 1: Figure S1, where the healthy movement has one submovement while the stroke affected movement has three submovements with a finite temporal gap between two successive submovements. A simulated scenario was used to investigate the validity of different existing smooth measures in [4], where simulated movements were generated by systematically varying the number of submovements and the inter-submovement interval.

The current neurorehabilitation and motor control literature reports several different measures for estimating smoothness of a given discrete movement [4, 14‐16, 36, 39, 40]; these are are listed in Table 1 along with the details of their different properties. The most common measures are the jerk-based measures [14], most of which are not valid measures of smoothness. The dimensionless jerk (DLJ) and the log dimensionless jerk (LDLJ), defined below, are the only valid jerk-based measures of movement smoothness [4]:

where v(t) is the movement speed, t is time, t ₁,t ₂ are the start and end times of the movement, and \(v_{\textit {peak}} \triangleq \max _{t \in \left [t_{1}, t_{2}\right ]}v(t)\) is the peak speed. DLJ lacks sensitivity in the physiological range [4], which the LDLJ addresses through the natural log function. However, both DLJ and LDLJ are very sensitive to measurement noise and have poor reliability [4].

The other most commonly used smoothness measure is the number of peaks (NP) measure that counts the number of maxima in a given speed profile v(t):

where \(\left |\cdot \right |\) represents the cardinality of a set. NP is a simple measure, but lacks sensitivity and robustness [4].

The spectral arc length measure (SAL) introduced in [4] uses a novel approach to estimate smoothness. It estimates smoothness by calculating the arc length of the Fourier magnitude spectrum within the frequency range 0 to 20 Hz of a given speed profile v(t):

where V(ω) is the Fourier magnitude spectrum v(t), \(\hat {V}(\omega)\) is the normalized magnitude spectrum, normalized with respect to the DC magnitude V(0), and ω _c is fixed to be 40π (corresponding to 20 Hz).

In this paper, we introduce a slightly modified version of the original SAL, which we call SPARC for SPectral ARC length, by setting

The SPARC extends SAL in that ω _c is adaptively selected based on a given threshold \(\overline {V}\) and is upper-bound by \(\omega _{c}^{max}\). In contrast to SAL, SPARC is independent of temporal movement scaling, and retains the good sensitivity and reliability of the SAL. A detailed explanation of the SPARC measure and the rationale for its development is given in Appendix A of the Additional file 2. The entire analysis presented in the paper was carried out using iPython [41]. The entire analysis can be found online at https://​github.​com/​siva82kb/​SPARC.

A general measure of movement smoothness is one that can be applied to any movement type, e.g. discrete or rhythmic; and on any movement-related data, e.g. kinematics, forces, impedance etc. Let us consider a general measure of movement smoothness λ _s where T in Eq. (1) is any type of sensorimotor task. Let x(t) represent the measured information associated with a motor action M _m that is generated in response to the given task T: where x(t) is the measured movement related variable, x _i (t) is the i ^{t h} component of x(t), and t represents time. The information contained in x(t) can be movement kinematics, force or even mechanical impedance, in either task or joint space, depending on the task and the sensing modality used for measuring the motor behaviour. For example, movement kinematics could be simple spatial or joint space location measured through a motion tracking system, or linear accelerations and angular rates measured using an inertial measurement unit. In the case of isometric tasks, it would contain force or torque in the task or joint space.

How does one analyse the smoothness of any type of task? The simplest method would be to apply an existing smoothness measure on x(t) independent of the task type. However, this approach does not work, as we demonstrate through a simple example involving a rhythmic movement. Let us consider a simulated experiment where three subjects (two experts and one novice) were instructed to move back-and-forth 10 times between two spatial targets P and Q. The subjects were asked to move at a comfortable self-selected speed, and rest at the targets for a short duration. The movements M _a , M _b and M _c made by the three subjects are shown in Additional file 3: Figure S2. The smoothness estimates obtained by simply applying the SPARC and LDLJ measures to the entire movement data are also shown in these plots in Additional file 3: Figure S2.

By visual inspection of the position data in Additional file 3: Figure S2, one immediately concludes that the movement M _b is less smooth than M _a and M _c , as M _b is more intermittent during some individual movements between P and Q. The smoothness estimates shown in Additional file 3: Figure S2 agree with this conclusion, as M _b is less smooth than M _a and M _c using both SPARC and LDLJ. The movement time (MT) and dwell-time (DT) of M _a and M _b are exactly the same. Thus, differences between their smoothness estimates is due to differences in intermittency of the individual movement components (movements between P and Q) of M _a and M _b .

Now, by applying the same logic, can we say anything about the smoothness of M _a and M _c ? Is one smoother than the other, or are they both equally smooth? Using the same argument as before, we would conclude that M _a and M _c are equally smooth, because in both M _a and M _c , the movements between the targets P and Q appear similar (except in M _c the movements are a little faster, the dwell-time is shorter). The results in Additional file 3: Figure S2, however, do not agree with our intuition in this case; smoothness of M _a and M _c are close but not equal. This difference is due to the sensitivity of SPARC and LDLJ to the temporal organisation of the individual movement components in the entire rhythmic movement (i.e. the relative values of MT and DT). When the SPARC and LDLJ are applied on an entire rhythmic movement, it is seen as one long discrete movement, with the temporal organisation of the individual movement components appearing as intermittencies, which affects the overall smoothness. This implies that changes in the number of components will also result in a change of the smoothness estimate. For example, consider another movement M _d (plot not shown) which has the same MT and DT as M _a but has 20 back and forth movements between P and Q instead of 10 in M _a . Here, the smoothness of M _a is higher than that of M _d (value shown in the bottom plot in Additional file 3: Figure S2), when in fact the smoothness of M _d is no different from that of M _a and M _c . Therefore, the conclusion is that the smoothness measures cannot be used on an entire rhythmic movement to estimate smoothness.

Consider a movement represented by x(t), \(t \in \left [t_{s}, t_{e}\right ]\), where t _s and t _e are the start and end times of the movement, respectively. This movement x(t) can be represented as a concatenation of a set of individual distinct (non-overlapping) movement components² x ⁱ (t):

where the i ^{t h} distinct movement component x ⁱ (t) is the product of the x(t) and the rectangular window Π _i (t); N is the number of distinct components in x(t), and t _i is the starting time of the i ^{t h} movement component, where t ₁=t _s and t _N+1=t _e , and \(t_{i} < t_{i+1}, \,\forall i \in \left \{1,2,3,\cdots,N\right \}\). We note that the representation in Eq. (6) is not unique, and one can choose the value of N and the t _i s arbitrarily. For movements M _a and M _b in Additional file 3: Figure S2, one possible representation in terms of Eq. (6) would consist of N=20 and \(t_{i}=1.25\left (i-1\right)\). This particular choice of values for N and t _i for M _a and M _b would segment out the individual discrete movements between the targets P and Q, along with postures at one of the targets. One can come up with a similar model for M _c with the parameters N=20 and \(t_{i}=0.7\left (i-1\right)\). The representation in Eq. (6) is a form of windowing procedure to segment the long movement into meaningful components for which the smoothness can be estimated individually; we call this the event-based segmentation procedure. With this procedure one can estimate the overall smoothness of a given movement by first estimating the smoothness of its individual components, and then combining the individual estimates to obtain a single number representing the “overall” smoothness Λ of the movement x(t):

where f(·) is a mathematical function that combines the smoothness estimates of the individual movement components in x(t), λ _i is the smoothness estimate of the i ^{t h} movement component obtained using the selected measure e.g. SPARC. The choice of function for f(·) is crucial, and this must be a function that ensures that the overall smoothness Λ satisfies some of the following properties:

Based on these properties, a good function for estimating the overall smoothness of a given movement from its components is the weighted average function:

where w _i ≥0 is the weight given to the smoothness of the i ^{t h} movement component λ _i , and the denominator in Eq. 8 is a normalising factor ensuring that property 3 is satisfied.

A nice feature of the weighted average scheme is that it can be used to summarise the smoothness of an entire movement or just specific parts of a movement. For example, consider the movement M _a in Additional file 3: Figure S2. Let us decompose this movement using Eq. (6) with parameters N=20, and t _i =1.25(i−1). Now, to estimate the overall movement smoothness one could choose \(w_{i} = 1 \,\, \forall i \in \left \{1,2,\cdots N\right \}\). But if we only wanted the overall estimate of all movements from P to Q in M _a (i.e. excluding movement from Q to P), then one could choose w _i =1 when i is odd and 0 otherwise. Moreover, the event-based segmentation scheme for smoothness estimation of movements also allows one to track the smoothness over the course of movement, on an event-by-event basis.

An important point to note about the general smoothness measure of Eqs. (6) and (8) is that the results of smoothness analysis strongly depend on the movement segmentation step, the details of which are controlled by the parameters N and t _i in Eq. (6). Although, technically they could be chosen arbitrarily, one must ensure that the choice of parameters N and t _i yields meaningful segmentation of the movement data. That is, the movement components should be clearly identified as specific events in the overall movement, or they could be specific events of interest for the purpose of an analysis. In any analysis, one must first choose the events of interest, which will guide the choice of segmentation parameters and the subsequent smoothness estimation. For example, a meaningful segmentation for movements in Additional file 3: Figure S2 would be to segment out individual movements between P and Q. In general, the segmentation step can be guided by metadata obtained from, (a) knowledge about the task that was being performed (e.g. task type, target locations, via-points, movement reversals etc.); (b) annotations³ of the movement data recorded during data collection; and (c) also from the actual movement data⁴.

Some form of movement segmentation appears to be the most preferred method, in the current literature, for analysing smoothness of rhythmic movements. Nasseroleslami et al. [30] had segmented (almost periodic) rhythmic movements into individual cycles, which were averaged before estimating the smoothness of averaged cycles. Even though here the approach was similar to the proposed general smoothness measure, the smoothness estimation process was different from that of Eq. (8). On the other hand, Ao et al. [21] and the gait studies [24, 25, 27, 29] can be seen as instances of the general approach described in this section. Ao et al. segmented (almost periodic) rhythmic movements based on the desired trajectory that was displayed to their subjects to follow [21]. While, the gait studies analysing smoothness using the HR, segmented data based on heel strikes detected from foot switch data collected during gait [25, 29]. However, these studies did not evaluate an “overall” estimate of the entire movement, unlike the proposed general measure.

As a final step in the discussion of movement smoothness analysis, we would like to list recommendations for researchers and clinicians interested in analysing movement smoothness.

¹ Measurement noise is the cumulative effect of noise in the measurement system (sensors and instrumentation), changes in assessment protocol, and also other unknown or unaccounted sources.

² Note that such a model is used purely for the purpose of smoothness analysis, and it does not make any assumptions about the organising principles governing the observed movement.

³ Annotation could be movement metadata recorded during the data collection process. For example, in motor control experiments the computer program recording movement data and presenting audio-visual feedback during the experiment, would need to make decisions on when a subject has reached a target, or when he/she has stopped moving etc. This sort of information could be later used to segment the data for analysis.

⁴ Let us consider the example of a figure drawing task where a subject was asked to draw a large symbol of infinity on a planar surface using his/her arm, repeatedly N times. Here, the subject could have started from any point on the infinity symbol and drawn the shape repeatedly. This is an example of an almost periodic movement [28]. To segment this data into individual cycles representing a complete drawing of the symbol, one could define the points of segmentation as the times where the actual movement states (e.g. position, velocity, acceleration etc.) almost repeat themselves. In this case the segmentation is entirely based on the actual movement data.