Inter-joint coordination is a topic that has attracted an increasing number of studies in recent years (Fig. 3). When searching for “upper-limb inter-joint coordination” in Google Scholar, 2450 articles were presented, 801 of them were published after 2018 and 319 after 2021. Nevertheless, less than one third of these used metrics based on kinematic data from discrete movements. A total of 12 metrics not including variants were drawn from existing literature and provide diverse methodologies for assessing inter-joint coordination. Certain metrics concentrate solely on particular time events, while others compare the complete trajectories of joints, and others examine overall conditions without delving extensively into joint specifics. This assortment of metrics can potentially emphasize distinct facets of inter-joint coordination. Of the 12 metrics used to measure inter-joint coordination, only two metrics are used in the majority of cases: Angle-Angle Cyclograms and Continuous Relative Phase (Fig. 4).

Fig. 3

Distribution in time of the articles considered in this review, showing an acceleration of the publication on the subject

Fig. 4

Distribution of the different metrics used in the articles considered in this review

In the next sections, metrics are presented by order of use, and the following convention is used:

\(\theta \in \mathbb ^n\) where n is the degree of freedom represents the joints trajectories,

\(\dot \in \mathbb ^n\) represents the joints velocities,

\(t \in \mathbb ^\) is the time,

\(\dot \in \mathbb ^3\) is the end-effector velocity

Each metric is illustrated with a theoretical example.

Continuous relative phase

Definition CRP, alternatively known as “Temporal Coordination Index” (TCI) [50], is one of the most applied metrics. CRP has been used to quantify inter-joint coordination since 1993 [51], firstly for the lower-limb, and only since the 2000s for the upper-arm [52].CRP extracts the phase angle of the relation between position \(\theta\) and velocity \(\omega\) for each joint on the overall movement, and then compares the obtained phase angle \(\phi\) between different joints. As suggested by [51], “Quantification of inter-joint coordination through the use of the relative phase angle provides information that cannot be obtained through conventional angular position vs time presentation and may lead to substantive differences in interpretation of kinematic data.” CRP transforms the data into a phase plane, enhancing the phase relationship between joints and enabling the determination of which joint takes the lead over the others.

CRP was historically referred to as “Relative Phase Angle” due to its computation involving the disparity between two phase angle signals. The term ”CRP” was introduced as a counterpart to “Discrete Relative Phase” (DRP), which calculates the phase angle between two joints at a specific instant in time, rather than tracking their evolution over the entire movement. DRP has often been used in gait analysis, leveraging readily identifiable time events such as foot placement [53]. However, such events are somewhat more challenging to capture in discrete upper-limb movements, and fail to provide a broader perspective of the entire motion. For these reasons, only CRP is used in this study.

CRP computation is based on the phase angle, \(\phi\) which is the angle between the position/velocity point and the null velocity axis (Fig. 5) of the normalized data.

$$\begin \phi _i(t) = tan^\left( \frac\right) . \end$$

(1)

Other techniques such as Hilbert transform can be used on sinusoidal signals to extract the phase angle of a dataset [54].

To compare 2 joints i and j their respective phase angles are subtracted (Fig. 5) as follows with i being the proximal joint, and j being the distal joint:

$$\begin CRP_(t) = \phi _i(t) - \phi _j(t), \end$$

(2)

Fig. 5

Continuous Relative Phase Computation. For 2 consecutive joints i (in blue) and j (in orange), position (first row) and velocity (second row) are computed and then plotted together (third row). Phase angles are extracted (fourth row) and subtracted (fifth row)

Analysis A constant relative phase means that the relation between the two joints is constant, while a positive or negative CRP means that one joint progressively goes out of phase with respect to the other joint (e.g. a negative CRP means that the proximal joint is lagging behind the distal joint and a positive CRP means that the proximal joint is leading the distal joint).

As CRP is a time series, it can be difficult to visually recognize differences between 2 CRP. In order to simplify CRP analysis, some studies focus on the Mean Absolute Relative Phase (MARP) and Deviation Phase (DP, equivalent to the standard deviation) [55]. It is proposed that if \(MARP<30 ^\) the joints are moving in phase. On the other hand, if \(MARP > 30^\) movement is considered as out-of-phase. A small DP indicates that the relation between the 2 joints is stable.

Other articles have suggested different tools to analyze the CRP pattern, such as linear regression [56]. Some variants of the CRP have also been used as Discrete Relative Phase (DRP), to evaluate the timing of key events in each of the angle profiles [53, 57]. However, such methods are more suitable for cyclic tasks since they compares the temporal dispersion of events.

Angle–angle plot or cyclograms

Definition The angle-angle plot is one of the oldest inter-joint coordination metrics [58]. Also named angular covariation plots [59] or cyclograms [60]. Angle-angle plots are 2D or 3D plots [61] where each axis of the plot is one joint’s position or velocity (Fig. 6). Angle-angle plots emphasis the relative trends in angular displacement for each joint over time [58]. By exposing those relative patterns, alterations in inter-joint coordination can be visually discerned. In certain instances, polar angle plots have also been employed to appraise joint coordinations [62].

Fig. 6

Angle-Angle Plot or Cyclogram. Two joints’ position (blue and orange) are plotted together (in green)

Analysis If a coordination exists between the 2 variables, a relation should appear as a distribution of data points around the diagonal line or plane of the plot.

This simple representation for coordination has been widely used, but characterizing (geometrically) the coordination pattern can be quite difficult due to the large number of curves to analyze (number of curves grows quadratically (\(n^2\))). In order to simplify the analysis of those plots, different characteristics can be extracted as: the global slope [63], the mean magnitude (mean distance between two consecutive points [56]), angular coefficient of correspondence (slope of two consecutive points [64, 65]) etc. Those measurements are also used as inter-joints coordination metrics.

Principal component analysis

Definition Principal Components Analysis (PCA) is a method used to reduce dimensionality of a dataset. First developed in the early 20th century [66, 67], the goal is to extract the main components of a multidimensional dataset, so it becomes easier to interpret since only independent features are left. In human movement analysis, PCA is used to extract the main independent relations between a set of joint trajectories.

PCA is computed on position or velocity-centered data. From the covariance matrix of the dataset, eigenvalues and the corresponding eigenvectors are extracted. Each eigenvalue represents the amount of variance of its corresponding component, while the corresponding eigenvector indicates how much each variable is contributing to the component. PCA can be computed both from the correlation matrix and from the covariance matrix. However, computing PCA on non-standardized data using the covariance matrix will remove data with a smaller range since they have less variability than data with a greater range. Alternatively, using the correlation matrix is equivalent to directly standardising the dataset (mean data is set to 0 and the standard deviation is set to 1).

Extended versions of PCA such as functional Principal Components Analysis (fPCA) have also been explored [68] in order to consider temporal variation in the movement (and not only postures, as is typically the case in PCA). Sparse Principal Component (SPCA) [69] is yet another variation of the PCA method that sets to 0 the variance of the variable that has a small variance, so they are not disturbing the interpretation of the main components. Since PCA is a global method, it doesn’t require precise joint-angle measurements to be able to extract main components of the coordination strategy used. For example, [70] directly used the 3D coordinates of markers placed on the human body during walking as inputs.

Fig. 7

Principal Component Analysis. Two joint positions (in blue and orange) are plotted together (in gray) in order to find a new vector basis (in green) that maximizes the variance of the dataset

Analysis PCA aims at extracting a common pattern in a dataset. The first step in the analysis of PCA results is to examine the percentage of variance explained by each Principal component (PC). The first component of the PCA represents the dimension of the data that has the largest possible variance in the data set, whereas the last components represent the data that have the smaller variance. In the example Fig. 7, the first PC accounts for 90% of the variance. Then, inside each PC, the weight of each original variable can be examined [20]. Here, the first PC is mainly composed of the position of the ith joint. Other PC analysis techniques have been developed. [71] has reconstructed the PC evolution over time by summing all the variables weighted by their PC results. This PC evolution has been analyzed in [72], using a hierarchical classification algorithm to detect changes in coordination. However, this last method works with a dataset where the first PC contains more than 90% of the dataset variance. When the first PC represents less variance, the analysis must be extended to the other PC, increasing the dimensionality of the results.

When comparing different PCA results, one hypothesis is to say that the alteration of inter-joint coordination leads to different joint contributions. A loss of inter-joint coordination leads to a higher number of PCs to explain the variability of the movement, since joints are uncoupled. Another indicator to check is the weight of each variable inside each PC. Indeed, a loss of inter-joint coordination leads to more simple and less variable contributions. For example, fewer PCs are needed to reconstruct the movements of stroke patients than non-stroke patients [68].

One main point of attention when using PCA is the size of the dataset. Indeed, when a too small dataset is used, the results obtained with the PCA are unstable. There is no ’minimal size’ for a dataset on PCA, one way to verify that PCA can effectively be used in this case is to bootstrap or cross-validate the PCA by deleting or exchanging some data in the dataset and computing the PCA again. If the result is similar to the result obtained with the original dataset, the result is stable, PCA can be effectively used in this case. If results vary a lot with the modification of the dataset, the result is unstable and should be used cautiously.

Cross correlation

Definition Cross Correlation is a signal processing method used to measure similarity of two series as a function of the displacement of one relative to the other (sliding dot product). When cross-correlation is used to measure coordination, usually only a zero time lag is considered [73], however recent studies have begun to explore the result of cross-correlation with different lags [74]. Cross-correlation is a metric that can be used on different types of temporal signals, for example [75] have computed cross-correlation based on CRP signals.

$$\begin corr_(k) = \sum _^ \theta _i(m) \theta _j(k-m). \end$$

(3)

Analysis The higher the cross-correlation is, the more the 2 joints are coordinated since their signal is considered similar. However, in case of a large difference in movement amplitude, the cross-correlation can be high even if the joints are poorly coordinated. Joint position may be normalized to avoid this artifact.

Atypical kinematics

Definition Atypical kinematics is a metric suggested by [76]. The goal of atypical kinematics is to extract portions of time normalized movement sets where joints trajectories differ from a reference movement using a filter based on PCA. The result gives a normalized time period and the joint sets for which the trajectories differed most.

Atypical kinematics needs to be computed in three distinct steps. The first step is to build a typical movement filter. The typical movement filter is built from the principal components extracted for each percent of the reference movement (from 0% to 100%). The second step is to filter all the data using this filter. To achieve this second step, each percent of the data to analyze are multiplied by the corresponding typical movement filter and then by the transposed of this filter. Finally, the last step is to define which part of the moment is atypical or not. To this purpose, error between the original and filtered data is computed for each percent of the movement. If the error is greater than 3 times the standard deviation of the reference movement, the movement is classified as atypical for this time period (Fig. 8).

Fig. 8

Atypical Kinematics. First Step (in orange), compute typical movement filter. Second step (in blue), filter the data. Third step (in purple), compute the error between the original and the filtered data

Analysis The more atypical kinematics moments there are, the less the movement is coordinated [77]. The same considerations regarding the size of the dataset in PCA should be applied for atypical kinematics.

Inter-joint coupling interval

Definition Inter-joint Coupling Interval (ICI) is a temporal metric used by [78]. This metric highlights relations between the end of the activation for each joint, also named settling time. This temporal delay provides insight into the sequence of joint deactivation and the duration for which one joint remains active after the cessation of movement in another. As is the case with many temporal metrics, ICI doesn’t provide comprehensive inter-joint coordination information spanning the entire movement but rather focuses solely on the distinct event of joint deactivation.

The joint’s deactivation is defined as the moment where the joint’s velocity is lower than 5% of its maximum absolute velocity. For each pair of joints, their respective end of deactivation time is subtracted (Fig. 9).

$$\begin ICI_ = t_ - t_. \end$$

(4)

Fig. 9

Inter-joint Coupling Interval. Inter-joint coupling interval is the delay between 2 joint’s settling (\(t_s\))

Analysis If ICI is close to zero, and the standard deviation of ICI for all movement is low, that means that the 2 joints are well coordinated. The higher the ICI value is, and the higher the standard deviation of ICI is, the less the joints are coordinated in a temporal manner.

It should be noted that even for healthy subjects, the average ICI might be slightly lower than zero, as a proximal-to-distal sequence of joint activation is generally observed [28] in routine tasks.

Distance between PC

Definition This metric, developed by [79], computes the angle between subspaces defined by Principal Components Analysis of the kinematic data obtained in 2 different conditions. The result of this metric quantifies the extent of disparity between two coordination strategies, rated on a scale from 0 to 1. However, details regarding this disparity are somewhat limited, as the specific joints with differing trajectories or the precise segment of movement exhibiting differences is not identified (Fig. 10).

Let U and V \(\in \mathbb ^\) be 2 subspaces defined by PCs of condition 1 and condition 2. The distance between U and V is defined as

$$\begin dist(U,V) = \sqrt^2(U^TV)}. \end$$

(5)

With U and V being a set of unit vectors defining a subspace and \(s_\) the minimal singular value of \(U^TV\).

Fig. 10

PCA distance. The distance between 2 subspaces defined by principal component analysis (in blue and red) is defined by the angle \(\Phi\) between the 2 spaces

Analysis The higher the distance is, the more different the two coordination strategies are [80]. However, this metric does not tell which joint of the coordination or part of the movement is coordinated differently.

Correlation coefficient statistics

Definition The Pearson Correlation Coefficient [81] or Spearman Rank Coefficient [82] are statistical tools used to evaluate the strength of a linear or monotonic relation between two variables. The correlation between two joints offers insights into the extent to which these joints exhibit coordinated behavior, indicating whether they evolve together in a linear or monotonic manner. The result ranges between \(-1\) and \(+1\) and a p-value is associated to the result. A small p-value indicates that a significant correlation exists between the 2 variables.

Analysis

The closer the Correlation Coefficient is to 1 or \(-1\), the stronger the linear relationship between the two joint angles. This result can be used only if the corresponding p-value is lower than 0.05 (5% significance level). If data are not filtered, strong correlation can emerge from noise inherent to the signals. A strong correlation can also be found if one joint moves only to a limited degree (due to the movement of the other joints or micro-movements for example), even if this behaviour might not fulfil the qualities expected of “coordinated movement”.

Angle ratio

Definition The Angle Ratio is another relatively simple measure that takes 2 joint trajectories, and for each timestamp computes the ratio of both angles. To simplify, only the ratio at the end-effector maximum velocity can be computed [83]. The angle ratio reveals which joint within a pair contributes the most to the overall movement. Yet, this metric’s accuracy might be influenced by joints with varying overall ranges. As an illustration, the wrist commonly exhibits a smaller angle measurement compared to the shoulder.

$$\begin AR_(t) = \frac. \end$$

(6)

Analysis If the range of motion of the 2 joints are different (i.e, wrist and shoulder), the data should be normalized at their range before computing the angle ratio. Between 2 conditions, if the joint angle ratio is increasing, that means that joint i is making a greater contribution to the movement. On the contrary, if the joint angle ratio decreases between 2 conditions that means that joint j is contributing the most.

Relative joint angle correlation

Definition Relative Joint Angle Correlation is a metric presented by [84] based on the analysis of the covariance matrix between 2 relative joint angles. Relative joint angles are calculated from the segment’s vector, this indicates that the kinematic parametrization of the chain used to extract joint angles (i.e, International Society of Biomechanics Convention for upper-limb [85], robot kinematic chain, etc.) does not affect the result. This metric in an indirect measure of inter-joint coordination based on kinematic data.

RJAC is a measurement of inter-limb synergy, and by extension the inter-limb synergies reflect the inter-joint coordination strategy. Notably, RJAC values have been demonstrated to exhibit correlation with the Fugl-Meyer Assessment of Sensorimotor Recovery After Stroke for Upper-Limb Extremity (FMA-UE) [84] which gauges motor abilities and joint amplitude. This association underscores how RJAC not only reflects motor skills but also provides insights into joint coordination.

Let i be the proximal segment and j the distal segment, defined as 3D vectors for each t timestamp (Fig. 11).

Fig. 11

Relative Joint Angle Correlation. RJAC is based on joint’s position \(\Delta\) between consecutive timestamp

\(\Delta \theta _j(t) = arccos\left(\frac(t). \textbf(t)}(t)||\textbf(t)|}\right) - arccos\left(\frac(t-1). \textbf(t-1)}(t-1)||\textbf(t-1)|}\right)\).

For the first joint, where there is no proximal segment, the relative angle is computed as follows:

\(\Delta \theta _i (t) = arccos\left(\frac(t-1). \textbf(t)}||\textbf(t)|}\right)\).

Then RJAC is computed as:

\(RJAC(i,j) = \frac}\).

Analysis To compare different datasets together, the absolute value of the RJAC matrix is considered. A lower value of the RJAC’s matrix determinant means that joints are less coordinated. Similar to other metrics that give only one final value for a whole condition, this metric provides insight into how different two datasets are. It does not provide insight on the origin of the difference.

Temporal coordination

Definition Temporal Coordination is used by [86] and focuses on determining the delay between the activation time of the joints compared to the beginning of the movement. Similar to the inter-joint coupling interval (ICI) metric, this measure focuses on a single time delay between two specific time events. Here, it emphasizes the point at which a joint begins to engage in the movement. Among individuals without impairment, this delay is typically minimal and follows a proximal to distal sequence, indicating that proximal joints usually initiate movement before distal joints, showing effective coordination across all joints.

Joint activation is defined as the instant when joint velocity is greater than 5% of its maximum absolute velocity (Fig. 12).

$$\begin TC(i) = t_-t_. \end$$

(7)

Fig. 12

Temporal Coordination and Zero Crossing Time Interval. Temporal Coordination is the delay between the beginning of the movement (\(t_\)) and the start of the joint (\(t_\)). Zero-Crossing Time Delay is the delay between the start of the movement and the end of activation of the joint (\(t_\))

Analysis As per the ICI, the smaller the TC is, the more the joints are coordinated from a temporal point of view.

Zero crossing time interval

Definition Zero-Crossing Time Interval is a metric that computes the time between the beginning of the movement and the deactivation of the joint [87]. It is also a temporal metric, focusing exclusively on the temporal synchro