2.1 Weighted Inverse Kinematics formulation

In order to control a robotic manipulator, one needs to deal with Forward and Inverse Kinematics. Forward kinematics refers to the problem of determining the end-effector position when the joint values are known. This is often a simple operation when the geometrical properties of the chain are known. Inverse kinematics, however, aims at finding the joint values based on a given end-effector position. This problem becomes more challenging in redundant systems; i.e., having more degrees of freedom than the dimension of the task. In this case, one has to choose one particular IK solution among infinite possibilities. The solution to the IK problem can be categorized into two groups: closed-form and numerical solutions [22]. Closed-form solutions rely on geometrical properties of the kinematic chain or solve the IK problem in an algebraic form. However, numerical solutions are robot-independent but rely on different heuristics or iterative techniques. Furthermore, the IK problem can be addressed at the velocity level; i.e., finding the joint velocities based on a given end-effector velocity. In this case, it is helpful to differentiate the forward kinematics to obtain the Jacobian matrix, which maps joint velocities to the end-effector velocities at a given joint configuration. Thus, the pseudo-inverse of the Jacobian matrix can be used to solve the IK problem as initially formulated in [23]. A tremendous amount of research has been dedicated to overcoming this approach’s main limitation: dealing with singular configurations leading to a non-invertible Jacobian matrix or near-singular configurations resulting in large velocities. Several approaches to mitigate this problem can be found in the literature: cyclic coordinate descent [24], Levenberg-Marquardt damped least square [25], quasi-Newton and conjugate gradient [26], neural networks and artificial intelligence [27–29], Fuzzy inference [30], Genetic algorithms [31], Adaptive control using Jacobian transpose [32, 33], singular value decomposition [34–36], Damped Least squared [25, 37, 38], Quadratic programming [39]. Furthermore, using weighted pseudo-inverse for redundancy resolution is a common practice for robotic applications. Using weights (to pick a specific solution that minimizes a cost) was initially suggested in 1969 [40] and later implemented to control robotic manipulators [41, 42]. Most often, the manipulator inertia matrix is used as the weight matrix to minimize the total kinetic energy expenditure of the robot [43–45]. Nevertheless, in many robotic applications, the weight matrix choice might not directly affect the task performance [46]. In the same line, it is rare to see such approaches employed to analyze human movement. For instance, in [47] IK weights were used for modeling the kinematic data, which are manually set. In summary, the weighted IK formulation is less appreciated in the literature since there is no systematic method to choose a set of appropriate weights. Thus, researchers implicitly pick the identity matrix, retreating to the right pseudo-inverse, which is satisfactory for control purposes (as in [48]) but not sufficient for analysis.

2.2 The use of Inverse Kinematics in human movement analysis

Numerous metrics for movement quality have been proposed in the literature to analyze human motion; see [49] for a review. In the current state of the art, the kinematic measurements are often limited to the mean and standard deviation of joint motions and correlations analysis across joints; e.g., shoulder-elbow correlation. In [50], such kinematic measurements were used to study inter-joint coordination in post-stroke patients and compared to the clinical evaluation of the impairment by FMA-UE (Fugl-Meyer Assessment for Upper Extremity [51]). Studies using robotic-assisted protocols show that stroke patients exhibit abnormal intralimb joint coupling, which is correlated with FMA-UE [52–55]. Despite such a consensus, these clinical measures cannot capture small changes nor distinguish behavioral restitution from compensation. Other methods to study inter-joint coordination have been tried in the literature. For instance in [56, 57], inter-joint coordination is considered as the temporal relationship between the joint values. Similarly, in [58, 59], “continuous relative phase” has been used to study the effect of fatigue on inter-joint coordination. Such studies show that while these metrics are beneficial for assessment, they depend highly on the movement tasks. This is due to the fact that inter-joint coordination is often understood as correlation across joints without correcting for the fact that they are working toward a goal. Model-based approaches can help overcome these issues in investigating and quantifying inter-joint coordination.

The pseudo-inverse of the Jacobian matrix, as a model-based approach, is extensively used in the literature to analyze human kinematic data, mainly from a “motor-control” point of view. For instance, the well-known “Uncontrolled manifold” (UCM) method uses the Jacobian pseudo-inverse (more precisely, the null-space projector) to decompose the joint velocities into task-space and null-space [60, 61]. Null-space velocities (also referred to as “self-motion” [62]) are those which do not create an end-effector motion. By comparing the variations in the two spaces, the UCM method computes how strong the null-space is controlled (compared to the task-space). Thus, UCM views inter-joint coordination as how strong the joints contribute to the control of the null-space. Nevertheless, the same model-based approach using the pseudo-inverse of the Jacobian matrix can be used to quantify other facets of the inter-joint coordination; i.e., how much each joint contributes to creating an end-effector velocity. However, using an “a priori” assumed pseudo-inverse (right pseudo-inverse in this case) among infinite possibilities (i.e., weighted pseudo-inverse [63]) is an ill-posed approach to model kinematic data. Regardless of the choice for the pseudo-inverse, the modeling error (i.e., the part that cannot be explained by the pseudo-inverse) is attributed to the null-space. In summary, it can be seen that the problem of understanding the inter-joint coordination is the problem of finding the appropriate pseudo-inverse; i.e., finding IK weights that fit the data and rely less on the null-space.

2.3 The applications of weighted Inverse Kinematics in assistive robotics

Robotic-assisted rehabilitation has proven to be as effective as conventional training for upper and lower limb motor movement [64–68]. Moreover, robotic interventions provide two primary advantages. First, these devices (compared to traditional methods) allow for reliable measurements. Such measurements can be used to better analyze and track patient performance throughout the therapy. One challenge in this approach is determining the anatomical joint values based on the robotic ones. Many methods try to solve this problem by exploiting the IK formulation (e.g., [69]), which lies outside the scope of this work. The second advantage of such robotic systems is their capacity to influence patients’ kinematic behavior. This influence can be active where the robot assists the motion or passive where the robot acts as a damper and imposes a viscous field; i.e., active or passive from the robotic point of view. For instance, in a previous work [70], we used an upper-limb exoskeleton to modify the joint coordination by applying a force field in asymptomatic participants. Similarly, methods such as Constraint Induced Movement Therapy (CIMT) have been proposed where functional joints are restricted in order to encourage and rehabilitate the dysfunctional ones. Other works such as [70, 71] aim to alter the subject’s synergistic behavior using wearable robots. Even though rehabilitation robotics allows for various therapeutical approaches, the analysis of patients’ movements in terms of inter-joint coordination is limited. The use of pseudo-inverse methods in rehabilitation robotics is often criticized for 1) providing multiple solutions, 2) generating unnatural postures, and 3) lack of solution when the Jacobian matrix is singular. Therefore, there is an inclination toward closed-form solutions for rehabilitative applications [72]; for example, see [73–80]. While these analytical models provide reliable IK solutions for rehabilitation purposes, they cannot be used to measure the quality of the motion or to explain human kinematic data and its variability; e.g., individual differences or the evolution of the subject’s performance in the course of the therapy. This begs for modeling approaches to kinematics data where there are free and interpretable parameters such as the weights in weighted pseudo-inverse of the Jacobian matrix.

2.4 From inter-joint to human-robot coordination

Besides robotic-assisted rehabilitation, the weighted pseudo-inverse approach is of particular interest for assistive robotics; especially with the emerging cobots, robotic prosthetics, and robotic supernumerary arms. The resulting knowledge about human inter-joint coordination helps roboticists to design better robotic systems; e.g., humanoid robots, which move, interact and assist in a human-like manner. More specifically, the new generation of assistive robots requires systematic and efficient tools for analyzing and designing human-robot coordination. This is a necessary step toward ergonomic and efficient execution of daily tasks such as reaching motion across all domains: rehabilitation (e.g., in stroke patients), restoration (e.g., prosthetic for individuals with amputation), and augmentation (e.g., supernumerary arms to enhance industrial operators). In our previous works [81, 82], we used the concept of compensatory behavior to propose a new control paradigm for prosthetic arms; i.e., “Compensation Cancellation Control” as a movement-based strategy in contrast with EMG-based methods. Furthermore, in a recent work [83], we proposed a prosthetic control strategy specifically using the weighted pseudo-inverse of the Jacobian matrix. In such scenarios, a metric for inter-joint coordination can help the designer to assess the performance of a robotic assistive device. In other words, one could quantify how much the robotic joint contributes to the final end-effector movement. This view can be extended to any leader-follower robotic system with kinematic redundancies when the objective is to achieve higher assistive performance; i.e., higher follower’s contribution. Supernumerary robotic arms are also examples of such systems [84–86] where the human users use their proximal joints to compensate for the lack of robot’s activity. In all these examples, we are facing a similar question: ““how do we know if the robot/follower contributes enough to the task and not the user/leader compensating for the lack of robotic activity?”. In this work, we show that the weighted IK formulation can quantitatively answer such questions about human-robot coordination.

2.5 Mathematical background and problem formulation

Let us consider the observation of joint velocities ( where n is the number of degrees of freedom) and end-effector velocities ( where m is task/end-effector space dimension). Due to the geometry of the kinematic chain, joint velocities are mapped onto the end-effector velocities as: (1) which is known as Forward Kinematics with as the Jacobian matrix. Let us note that the Jacobian matrix depends on the joint configuration (). However, the mapping from end-effector to joint velocities (i.e., IK) can be modeled as follows. (2) where JJ^# J = J and . It is trivial to show that the resulting from any pair of J^# and who satisfies these two conditions satisfies the forward kinematics in Eq 1. In this IK formulation, is the generalized inverse of J (also called Moore–Penrose inverse). Furthermore, represents the null-space velocities since it does not affect the end-effector velocities. Given a particular J^#, the corresponding null-space velocity can be computed as: (3) where N = I−J^# J is the null-space projector of J^#. A practical and general choice for the J^# is the weighted pseudo-inverse [63]: (4) where is a positive definite matrix. This can be seen as the solution to the following problem: (5) A diagonal W suffices in most applications as it already provides enough flexibility for design and analysis purposes. Moreover, multiplying W by a scalar does not affect Eq 4. Therefore, W can always be normalized by its largest element, leading to the following choice for this matrix: (6) Therefore, our goal is to estimate such weights (w₁…w_n) from the observed kinematic data (, , and J). However, it is essential to note that any choice of W can model the observed data. In other words, each choice of W brings us to a specific decomposition of observed joint velocities as follows (7) where and which can be written as follows as well. (8) (9) (10) (11) (12) To illustrate the result of decomposition under different assumptions of W, we consider the following simple example of a redundant robot made of two serial prismatic joints with a one-dimensional task; i.e., n = 2 and m = 1. Let us consider an observed with J = [1, 1] which leads to . Using W = I to consider similar cost/contribution for each joint, we have J^# = [.5,.5]^T, which leads to the following decomposition between task-space and null-space: (13) While using W = diag([1, 1/9]), we have J^# = [.1,.9]^T with the following decomposition: (14) The difference between the two cases reflects the underlying assumption about the IK strategy. In the first case, we assume that the two joints (being redundant) will contribute equally to the task (i.e., 0.5 each). However, the fact that the first joint has zero velocity is explained by the null-space; i.e., −.5 for the first joint and 0.5 for the second joint. In the second case, we assume a higher gain/cost for the first joint, which explains the observed velocity as mostly task-related () with comparatively less utilization of the null-space.

This simple example reflects our philosophy in modeling the joint velocities observed in a redundant system: The observed joint velocities need to be mostly explained by where we correct for the imbalances of the contributions of joints by using a proper weight matrix. As seen above, uniform joint contribution puts a strong assumption on IK strategy, which is not in line with the fact that joints in kinematically redundant systems have different roles, functionality, costs, and contribution. Therefore, by not correcting for such imbalances, we risk having a decomposition that is not descriptive of the actual underlying strategy for redundancy management. To summarize, it is important to make a distinction between the two following questions:

• How is the redundancy/abundancy utilized? (modeling the IK strategy)
• How is the geometrical null-space utilized? (velocity decomposition into task-space and null-space)

In our view, it is essential to begin by answering the first question by precisely modeling the kinematic data; e.g., in our case, estimating the IK weights. Only after answering the first question (i.e., finding the proper null-space projector) can it be examined how the null-space is utilized.

This work explains the observed kinematic behavior by estimating a proper set of IK weights. These weights can be interpreted as the relative cost of the joints as described in Eq 5; i.e., joints with lower/higher weights are less/more costly to move. These weights, therefore, explain inter-joint coordination in terms of joints’ contributions to creating the end-effector velocity.

3.1 Proposed algorithm for estimating Inverse Kinematics weights

Let us begin with a single observation at time-step k with as the joint velocities, as the end-effector velocities, and as the Jacobian matrix. We assume that this observation is generated by the following IK model: (15) where is the weighted pseudo-inverse of the Jacobian (J_k) with the diagonal weight matrix . In this model, v_k denotes the null-space velocities which do not affect the task space; i.e., J_k v_k = 0. However, our estimation process is formulated as follows. (16) where , , , and represent the estimations for their respective variables. To estimate the null-space velocities , we use the current estimation of the null-space projector; i.e., . However, we use 0 ≤ γ ≤ 1 as the “null-space projection ratio” to consider a portion of the projected velocities. The effect of γ becomes clear when inspecting the modeling error e_k which can be rewritten using Eq 16 and as: (17) which shows that there is no modeling error when entirely relying on the null-space projector; i.e., e_k = 0 for γ = 1. In other words, any weight matrix can perfectly model the data; for example, . This issue is further explained in the S1 File However, with smaller ratios, the error depends on the choice of . This allows us to determine a weight matrix that explains the underlying IK strategy rather than exploiting the null-space as a means for modeling error minimization. On the other hand, a small γ might lead to an undesirable estimation as it seeks to explain the null-space velocities using only the task-space part of the model.

Algorithm 1: Identification of W from observed data for n joints, m task dimensions, and K samples.

 input: , , and

 parameter: γ

 output:

1 Initialize and ;

2 while e > ϵ do

3  ;

4  e = 0;

5  for k ← 1,…, K do

6   ;

7   ;

8   ;

9   ;

10   ;

11  end

12  e ← e/K;

13  ;

14  ;

15  ;

16  Rearrange into ;

17  Rearrange into ;

18  ;

19  ;

20   Solve the SQP problem with H and F as in Eq 24;

21  ;

22 end

Algorithm 1 provides a heuristic to compute IK weights from such observed data. We assume that there are collected samples from a kinematic chain at each time-step k = 1,…, K which serve as the input to our algorithm; namely, the joint positions , end-effector velocities , and as the Jacobian matrix at time-step k = 1…K. We initialize with and zero null-space velocities where . In this algorithm, lines 1–9 correspond to Eq 16 where we also introduce an auxiliary variable since we can write (from Eq 16): (18) where we define for kth time-step and in the matrix form as . Therefore, we can write Eq 18 in the matrix form as: (19) where we have the unknown variables as the linear coefficients, even though U implicitly depends on W in a nonlinear fashion. Furthermore, we can multiply both sides with and reach (20) where and . In this formulation, both and depend on our current estimation (). However, we can use our current estimation of and and update our estimation for in a quasi-static manner as: (21) where denotes our estimation at tth iteration. Since is a diagonal matrix, we can arrange and regarding their columns as follows: (22) where where is the j^th column of . similarly, , and . In the S1 File, we show that Eq 22 is equivalent to quasi-newton methods.

At this stage, one might consider using least-square methods to solve for , but there is no guarantee that the resulting values are positive. For this reason, we use Quadratic Programming in order to satisfy such constraints. To do this, we consider the following quadratic cost: (23) which can be simplified into: (24) where and . We solve the QP with lower and upper bound constraints for the solution; i.e., to have each weight between zero and one. Finally, we normalize the weights in order to have the biggest weight at 1.

Our algorithm has two parts: evaluation (lines 2–11) and update (lines 12–20). While possible to swap the two parts, having the evaluation first allows us to have the performance of our initial guess for W. Since, in this work we start with W = I, we have the comparison of our result with the conventional W = I used in the literature. Finally, we stop the algorithm when the average norm-2 error is smaller than a designated threshold (line 2).

3.2 Experimental and numerical validation

To validate our proposed algorithm, first, we consider two low-dimensional simulations. The main purpose of these simulations is to illustrate the overall behavior of the proposed algorithm in terms of convergence to the nominal IK weights. The further details and the results of these simulations are reported in Subsection 4.1.

For the experimental validation, we consider kinematic data recorded in two different scenarios in the context of robotic-assisted rehabilitation and upper-limb prosthetic robotics, respectively. These studies were approved by the ethics committee of the Paris Descartes University (CERES—IRB number 20162000001072) and the Sorbonne University (SU CER-2021–111), respectively. All participants gave written and informed consent before participation. Further details and the results of these investigations are presented in Subsections 4.2 and 4.3.

4.1 Illustrative examples

In the first example, we provide a numerical evaluation in which all inputs to our algorithm are randomly generated (i.e., independent and identically distributed) using normal distributions. This means that all elements in and J_k are sampled from while V is sampled from which allows us to study the effect of null-space velocity later. We consider n = 5 and m = 3 with K = 500 data points with w* = [1, 0.8, 0.6, 0.4, 0.2]^T. For QP-solver, we set optimality, constraints, and step tolerances to 1e−3. We use γ = 0.6 and σ = 0.2. The results are illustrated in Fig 1 where the parameters converge to their respective optimal values. The final error is due to the null-space velocities. We show in the S1 File that a higher level of randomness in the null-space (σ) leads to a higher final error.

Fig 2 studies the effect of γ on the convergence behavior. These results suggest that γ should be kept as low as possible. However, we should note that in this simulation, the null-space velocities v_k and are uncorrelated (as they are generated in an iid manner). In the next simulation, we see that γ acts differently in practical and realistic scenarios since null-space and joint velocities are correlated. Nevertheless, this simulation confirms that γ = 1 does not lead to any weight update as analytically investigated earlier.

In a second simulation, we consider a redundant kinematic chain with two joints and a one-dimensional task; i.e., n = 2 and m = 1. For the nominal IK solution, we consider w₁ = 1 and w₂ = 0.01 where we expect the task will be achieved by the second joint. Furthermore, for the null-space behavior, we consider v_k = [−0.1, q_k1, 0]^T where we expect the first joint to be at zero. We simulate this system for 6s with dt = 0.1 starting from the initial position of q₁ = [0.5, 0.5]^T. This provides us with K = 60 data points. For the desired end-effector position, we use x_g = 1, then at t = 2 we switch to 1.5 and at t = 3 to 0.5. The results of our algorithm are illustrated in Fig 3. In this case, having γ > 0.5 is necessary for satisfactory convergence of the parameters. It is also interesting to see that, when the null-space velocities are not considered (γ = 0), the algorithm mistakenly converges to w₁ = 0 and w₂ = 1; i.e., describing that the task is accomplished mainly by the first joint, and the resulting discrepancy (between observed and expected joint velocities) is due to the null-space velocities. On the other hand, choosing higher values for γ leads to convergence to the optimal weights but at a lower convergence speed. In this manner, γ = 1 can be seen as a particular case where the convergence time is infinite. One way to deal with this trade-off is to use a time-varying ratio. For example, in this simulation, we tested γ₀ = 0.6 with γ_{t+ 1} = γ_t+ 0.6(1−γ_t) in order to benefit from the fast convergence of γ = 6 at the beginning, and the optimality of a high γ toward the end. Generally, there is no procedure for choosing an optimal γ since the results depend on the dataset, as we see from the two last simulations. A helpful rule of thumb is to start with γ = 0.9; i.e., sacrificing convergence speed for optimality.

4.2 Experiment 1—Inter-joint coordination in a viscous field

We collected kinematic data from 17 asymptomatic participants (11 males and 6 females), reaching three different targets. Each target was reached in 20 trials (some trials had to be removed due to recording issues) under two conditions: “Natural” and “Viscous”. As shown in Fig 4, the participants were wearing a shoulder-elbow exoskeleton; i.e., 4-DOF ABLE upper-limb exoskeleton [87]. This device allows us to impose a viscous force field (as presented in our previous work [70]) to modify the subjects’ inter-joint coordination. The ABLE exoskeleton offers quasi-static gravity and friction compensation with highly reversible mechanical transmission. However, due to a lack of dynamic compensation, the device exhibits a certain level of undesired resistance which is characterized in our previous work [88]. Nevertheless, this factor remains a constant effect across the two conditions since the participants wore the exoskeleton during both conditions. Furthermore, the Wrist movements were blocked using a prefabricated orthosis. In this manner, the analysis is limited to a 4 DoF kinematic chain (matching the exoskeleton’s DOF where we can apply an arbitrary force field) performing a 3 DoF task, leaving a redundant DoF.

Participants performed pointing movements towards three targets (1: high, 2: forward, 3: inward) with adjusted distance and height for each participant. The final orientation was not specified nor constrained, leading to a 3-dimensional task. The joint rotations were expressed according to the ABLE kinematics:

• q₁: shoulder elevation around an anteroposterior axis.
• q₂: shoulder axial rotation.
• q₃: shoulder flexion in the plane defined by the two previous angles
• q₄: elbow rotation.

Positive values indicate elevation, internal rotation, shoulder flexion, and elbow flexion. Finally, the mechanical joint limits of the exoskeleton were never reached by the participants during the reaching movement.

Fig 5 shows the joint velocities of one of the participants reaching for the different targets. Here, we present the average velocities over trials, with the standard deviation as the shaded areas. Some observations can already be done at this stage. In the “Natural” Condition (i.e., transparent robot), the task is achieved mainly by utilizing the third and fourth joints (shoulder and elbow flexions), except for the third target, which requires the contribution of all joints. The main visible difference between “Natural” and “Viscous” is the higher utilization of the shoulder abduction. Moreover, these plots suggest that this participant has higher motor variability in the “Viscous” condition; i.e., larger shaded areas, especially for the first joint.

Fig 6 shows the result of our estimation algorithm for the IK weights of all the participants in the two conditions. In both conditions, the weights for the first and fourth joint are equal to one. As we presented earlier, we always scale the weights in order to have the largest value at 1. However, in this experimental setup, the fourth joint is over-specified; i.e., the effect of the fourth joint on end-effector velocity cannot be written as a linear combination of the other three joints. Therefore, for the IK, it does not matter what weight the fourth has. However, as a convention, we set this weight back to one right after solving QP, and we normalize only the first three first weights.

The estimated IK weights in the “Natural” condition show that, in a redundant configuration, the participants utilize the second and third joints over the first one. However, in the “Viscous” condition, the weight of the third joint significantly increases. This means the third joint is less used; consequently, other joints contribute more to the task. The resulting difference in IK weights (in Fig 6) provides a clear distinction between the two conditions; a distinction which is not easy to quantify from the trajectories shown in Fig 5, even though the qualitative difference is visible to the naked eye.

To go further, we can look at the extracted null-space velocity under the estimated IK weights. These velocities are illustrated in Fig 7 which is averaged over all participants for the first reaching target. Moreover, we compare our extracted null-space velocities with the case where the identity matrix is used; i.e., W = I assuming equal contributions for all joints. Our results show that a major portion of the data is already explained by estimating the proper weights; i.e., the extracted null-space velocities are smaller compared to the case with W = I. Moreover, the “Viscous” condition has a higher level of null-space velocities (with a higher variance). However, W = I shows the opposite; in the “Natural” condition, humans utilize their null-space more with a higher variation. These results show that analyses over the null-space velocities computed with inappropriate weights can be deceptive.

4.3 Experiment 2—Human-robot inter-joint coordination

In this section, we examine an experimental case where the goal is to restore a natural IK strategy (i.e., joint movement) using a robotic prosthetic arm with different control modes. The proposed IK weights identification method was used to analyze the IK strategy obtained with the different control modes and identify the one restoring natural IK strategy. To this end, we asked participants to control a 3-DOF planar kinematic chain shown on a display screen using their upper body as shown in Fig 8; i.e., the three degrees of freedom on the screen correspond to the rotation of the hip (q₁), shoulder (q₂), and elbow (q₃) in the sagittal plane. Using marker-clusters attached to the participants upper body, we tracked and transferred their body movements ([q₁, q₂, q₃]) to a virtual kinematic chain displayed on a screen (). While we directly mapped the hip and the shoulder, different mappings from the participant’s elbow to the virtual one () were used in each condition. Here, we detail these conditions:

• Natural: The elbow is directly mapped. Thus, the participant has full/direct control over the virtual kinematic chain; i.e., .
• Locked: The virtual elbow is locked at a specific angle; i.e., . Thus, the participant has partial control over the virtual chain.
• Coupled: The virtual elbow velocity is coupled to the participant shoulder velocity; . Thus, the participant has partial but indirect control over the virtual elbow.
• Assistive: The virtual elbow is controlled based on our previously proposed method in [83]. Thus, the participant has partial and indirect control.

In each condition, the participants reached for a sequence of target positions; i.e., 28 different targets, each displayed for 6s. We performed this experiment with 10 participants.

The controller used in the Assistive condition can be summarized as follows: (25) where and N₃ are the third rows of J^# and N, respectively. J^# is computed as in Eq 4 with W = diag([1, 1,.15]). To cancel the compensatory role of the hip and the shoulder, we use K = diag([2,.2, 0]), and rad. v can be considered as the low-pass filter version of the end-effector velocity with α = 40. In simple words, this proposed method amplifies at the end-effector level the velocities the human subject creates by the natural joints while helping the user with the posture. This controller can be seen as a nonlinear task-dependent synergy approach; i.e., a nonlinear mapping from and to . During this mode, the position of the virtual elbow (q₃) was limited to [0, π] rad.

The results for the estimated IK weights are presented in Fig 9. For the first condition (“Natural”), we estimate high weights for the hip and shoulder but a low one for the elbow. This means the human participants prefer to use the elbow joint even if the same end-effector velocity can be created using other joints. The result for the second condition (“Locked”) is straightforward. In this case, the virtual elbow is not contributing to the end-effector velocities, thus a high IK weight. Moreover, if the same end-effector velocity can be created by the hip or the shoulder, the participants use the shoulder; therefore, a lower weight for the shoulder. In the third condition (“Coupled”), we have a low IK weight for the hip. This shows that the participants tend to use the hip when it comes to hip-shoulder or hip-elbow redundancy. Furthermore, comparing the IK weight of the elbow to the shoulder reveals that such simple synergy methods are only effective locally; i.e., the elbow contributes more than the shoulder, while the hip compensates for the lack of contribution of both shoulder and elbow.

In the last condition (“Assistive”), we have satisfactory IK weights as the joints contribute more when moving up the kinematic chain (proximal to distal). Compared to the Natural case, we have a high weight for the hip (i.e., compensation cancellation) and low weight for the elbow (i.e., proactivity of the prosthetic joint). Thus, the main difference is the behavior of the shoulder joint. This means that in shoulder-elbow redundancy, the elbow is not fully taking over the shoulder as in the Natural condition. This result is expected given the nature of decentralized control in leader-follower setups; i.e., the natural joints need to initiate the desired end-effector velocity as a means for intention communication. In other words, the elbow joint cannot assist until it observes an end-effector velocity generated by the hip or the shoulder.

This result shows that our proposed estimation algorithm is capable of pinpointing the differences across conditions where different control strategies are used for a prosthetic joint. The resulting IK weights are interpretable (given the IK formulation) and coherent with the settings in each condition. Moreover, the resulting IK weight favors our assistive controller as it leads to a set of estimated IK weights that are qualitatively closer to the “Natural” condition. This result corroborates our previous findings where this type of assistive strategy leads to lower hip utilization [83]. Hip-utilization is an intuitive metric since the human participants tend to perform the 2D task mostly with the shoulder and elbow joints in the “Natural” condition. The participants begin to use the hip joint as compensation when they lose direct control over the virtual elbow joint; i.e., using the hip and the shoulder joint. Therefore, in this experiment, the inter-joint coordination can be interpreted as non-using the hip joint. In general cases, the objective of a prosthetic joint can be formulated similarly; i.e., obtaining a “naturalistic” inter-joint coordination or IK behavior which can be quantified using our proposed estimation algorithm. Finally, it is also interesting to note that we use weighted IK formulation in our proposed controller. As discussed in [83], these nominal weights characterize the dynamics and subsequently influence the system’s performance. However, it is important to note that the estimated weights are not only a result of the underlying control mechanism but also the executed motions.