Contribution—Motivation

The main contributions of this work can be summarized below:

• Identify the properties of kinematic redundancy when performing a specific task and establish the kinematic and dynamic relations (sections “Relation between task and joint space quantities” and “Task space equations of motion”). Determine the properties of dynamic redundancy and their relationship to joint and muscle space quantities both at a kinematic and dynamic level (sections “Relation between muscle and joint space quantities” and “Muscle space equations of motion”). This description can potentially enable the modeling and evaluation of hypotheses related to normal and pathological conditions presented in the coordination process (e.g., slack muscle disorder, increase in joint stiffness, rigidity, etc.). We also show that these disorders manifest themselves in the null space.
• Propose an alternative representation of the Equations of Motion (EoMs) in muscle space (section “Muscle space equations of motion”). Muscle space projection is a convenient representation for interfacing with segmental level models (e.g., reflexes) [3] or implementing controllers for tendon driven robots [4]. Furthermore, it has the highest number of DoFs and permits coordination of different aspects of the movement (e.g., co-contraction) that are uncontrollable in subspaces of lower dimensionality. Task and joint space projections are specializations of muscle space.
• Establish an effective model for quantifying redundancy with respect to the movement task, the neuromuscular model (e.g., linear/nonlinear muscle model, synergy encoding [5]) and the anatomical characteristics of the muscle routing (the muscle moment arm and its null space) (section “Exploitation of kinematic and dynamic redundancy”). This approach is advantageous as compared to the local techniques (e.g., minimum effort optimization [1]) that are suitable for finding only a particular solution, because it identifies the entire solution space of possible muscle force realizations as well as the factors that influence its structure. The importance of identifying the entire solution space when estimating the bounds of the joint reaction loads is demonstrated and we further show that misinterpretation of the results is possible if local techniques are used as compared to the proposed approach.

The results demonstrate working examples of the developed models. The projected EoMs in muscle space provide a new perspective for studying the musculoskeletal system in this domain. In section “Muscle space projection and reflexes”, we design a posture controller that encapsulates the characteristics of the internal regulation process performed in the spinal cord and study the response of the system under external disturbances. Muscle space projection proves to be a convenient representation to solve this problem, since no conversion to muscle activation is required.

In section “Characterization of the feasible muscle force space”, we present a case study in which we determine the force variability and correlations between different muscle groups for arbitrary movements and show that there is a significant systematic correlation between different muscle pairs that emerges from the muscle routing properties and the functional constraints of the task. This can be used to estimate the importance of actors/muscles during the movement, to identify synergies and classify the various control strategies available [6, 7], considering the entire solution space. This not only demonstrates an effective approach for finding the family of possible solutions, but also exposes the structural relations of the null space solutions.

In section “The effect of null space on the joint reaction loads”, we perform a joint reaction analysis on a gait, using a realistic full body model, appropriately accounting for the null space muscle forces that do not alter the movement. Accurate joint reaction force estimation requires reliable assessment of the muscle forces. As this is an impossible task, we demonstrate the usefulness of identifying the feasible muscle forces in order to determine the limits of the reaction loads. The large variance that is evident from the results confirms that misinterpretation of the reaction loads is possible if the null space contribution is ignored.

To the best of our knowledge this is the first study that incorporates the null space as shaped by the redundant nature of the system into the kinematic and dynamic relations. We investigate the properties of these projections with emphasis on their physiological counterparts. Thus, a complete theoretical analysis is constructed, which can be extended for many practical applications related to the motor coordination. The source code along with any related material for this publication are publicly available, providing simple examples so that the readers can reproduce, understand and reuse the presented methods (S1 File).

Related work

Motor control is the process by which humans and animals use their nervous system to activate and coordinate the muscles and limbs involved in the performance of a movement. The redundant nature of the musculoskeletal system was noted over fifty years ago [8], highlighting the over-availability of DoFs for most common tasks. Subsequently, researchers have created formal descriptions of the physiological process in order to identify the main variables responsible for coordinated movement [9–12]. Important hypotheses relative to these have been developed, improving our understanding of the central coordination problem [13, 14].

It has been shown that the musculoskeletal system can act as a “filter” that transforms the feasible muscle activations into a set of feasible wrenches [15, 16], revealing an important direction towards a holistic identification of the system capabilities. We propose a more general approach (section “Exploitation of kinematic and dynamic redundancy”), where the computed feasible muscle forces are both action-specific, accounting for the dynamic evolution of the motion, and satisfy the physiological constraints of the muscles, outlining the various factors that affect the solution space. Since the entire solution space is obtained using our framework, we can quantifiably explore the available strategies to the Central Nervous System (CNS). This is useful because it is still unclear how the CNS explores the musculoskeletal redundancy by controlling a low-dimensional subset of synergies [5, 17] and whether there is a correlation with the structure of the musculoskeletal system and the functional constraints each task [16, 18]. We will show that both the structure of the model and the task (movement) indirectly impose some degree of muscle correlation (section “Characterization of the feasible muscle force space”).

The use of linear projection operators have been successfully utilized in the field of robotics [19–21] in order to separate and simplify the control problem into multiple objectives. It has been shown that task-based projection can be used effectively for planning and simulation of constrained musculoskeletal systems [22, 23] and furthermore, it provides the means to identify the kinematically redundant DoFs. In a similar vein, we explore the null space properties as well as the potential applications that exploit not only the kinematic redundancy, but also the dynamic redundancy of the system. Detailed derivations are presented in order to provide the reader with the appropriate background to understand the implications of the proposed models for addressing redundant systems.

Relation between task and joint space quantities

This section focuses on the study of kinematic redundancy, which in turn will pave the way towards addressing the dynamic redundancy. Task space projection, a method that can reveal the properties of kinematic redundancy, has been thoroughly studied [19, 22] and we will briefly introduce the basic concept here. Establishing a link between task execution (e.g., movement of the hand) and muscle coordination is essential, as it is more convenient to interpret observations in task space rather than joint space. Furthermore, it is of great importance to transform the complex movement of the musculoskeletal system as a composition of multiple interleaved task goals, which may then be studied separately [24].

The task space position (x_t) is given as a function of the generalized coordinates (q) (8) implying that the generalized coordinates fully describe the task. A task has an abstract meaning, it can be interpreted either as a position, an orientation or a spatial primitive, which is a combination of position and orientation. The first derivative of Eq (8) (the dot notation depicts a derivative with respect to time) is given by (9) where the task Jacobian matrix () defines a mapping from joint to task space () and in most cases is a fat matrix. The inverse of Eq (9) must be augmented to account for the kinematic redundancy (Eq (7)) when mapping from task space (—low-dimensional) to joint space (—high-dimensional) (10) where represents the right null space of J_t (Definition 1) and an arbitrarily selected vector in . The derivative of Eqs (9) and (10) with respect to time are provided below (11) (12) where is also an arbitrarily selected vector in .

The task Jacobian defines a dual relation between motion and force quantities. The virtual work principle can be used to establish the link between task and joint space forces (13) where defines a mapping from a low- to high-dimensional space. With the addition of the null space forces, the relationship between task forces and manipulator joint space generalized forces takes the following general form (14) (15) where is an arbitrarily selected vector and Eq (15) the inverse mapping of Eq (14). Note that the null space complements the kinematic and dynamic relations in the sense that there are several combinations of joint space velocities, accelerations and generalized forces that bear no effect to the corresponding task space velocities, accelerations and forces. Furthermore, this redundancy is explored by the same null space term (), both at a motion and force level by a virtue of the MPP properties (Definition 2).

Remark 1. [Task motion—force duality]

The projection operators derived from and span the same subspaces.

Proof. Appendix “Task motion—force duality”.

Task space equations of motion

In the previous section (“Relation between task and joint space quantities”), the various relations between motion and force quantities of the kinetically redundant system were established. The current section continues with the derivation of the EoMs that relate the task acceleration to task space forces. Let the joint space EoMs have the following form (16) where denotes the symmetric, positive definite joint space inertia mass matrix, n the number of model DoFs and the joint space generalized coordinates and their derivatives with respect to time. The term represents the sum of all forces that act on the system (e.g., gravity, Coriolis, centrifugal, external, etc.), whereas the vector of applied generalized forces that actuate the model.

We can project Eq (16) onto the task space by multiplying both sides from the left with J_t M⁻¹ and using Eqs (11) and (14) (17) where denotes the task space inertia mass matrix, the task bias term and the generalized inverse transpose of J_t that is used to project joint space quantities in the task space.

At this point it is worth commenting on the different types of null space operators. If the null space is derived using the MPP (as in Eq (14)), then the projection operator does not ensure fully decoupled control in both task and null space. It has been shown that from the many projection operators that map on the null space [25], there exists a unique generalized inverse () that ensures this decoupling [19], which is a useful property in the presence of multiple prioritized tasks [26]. More specifically, suppose that (18) then we seek to find the generalized inverse that ensures zero contribution of the null space forces to the acceleration of a task (19) where the last requirement () implies that (20)

Remark 2. [Task space generalized inverse]

The null space projection operator cannot decouple task and null space forces, owing to the uniqueness of the generalized inverse Jacobian that achieves this decoupling.

Proof. [19].

Relation between muscle and joint space quantities

Interestingly, we can follow a similar approach by establishing the corresponding relations between the muscle and joint space quantities in order to address the dynamic redundancy problem.

The musculotendon length is given as a function of the generalized coordinates (q) (21) implying that the generalized coordinates fully define the musculotendon lengths, assuming that the muscles are pretensioned. The derivative of Eq (21) with respect to time is given by (22) where the muscle moment arm () is always a tall matrix. Due to the over-availability of muscles, Eq (22) can be reformulated to account for any change in that does not contribute to a change in (23) (24) where represents the null space of R, an arbitrarily selected vector in and Eq (24) the inverse mapping of Eq (23). The derivative of Eqs (23) and (24) with respect to time are given by (25) (26) with being an arbitrarily selected vector in . The null space contribution indicates that the CNS may regulate the lengthening/shortening of the muscles without altering the corresponding joint space quantities. This property can be used in coordinating different aspects of the tendon driven limbs or in the modeling of several pathological conditions of the muscle coordination process, such as slack muscle disorder.

From the virtual work principle (as in Eq (13)) we can establish a relationship between joint and muscle space forces (27) where . The negative sign convention is introduced, admitting that a muscle induces positive work during shortening (). The muscle forces f_m can be written as the sum of the two mutually orthogonal vectors and (Eq (7)). In contrast to Eq (27), the direct mapping does not require any reformulation, but the inverse requires reformulation in order to account for the redundancy effect (28) where is the same matrix defined in Eq (23), revealing the dual relationship between the subspaces of R and R^+T (Remark 3). Note, however, that this definition spans the entire , whereas in reality muscle forces are strictly positive (contraction) and bounded (limited force), which makes the above definition physiologically ill conditioned. In section “Exploitation of kinematic and dynamic redundancy”, we will provide an effective solution to this problem. If a solution satisfying the required torque exists, then there are infinitely many combinations of muscle forces () that do not alter the overall required torque. On a physiological level, this strategy is explored by the CNS in order to regulate joint stiffness through muscle co-contraction [7].

Remark 3. [Moment arm motion—force duality]

The projection operators of R defines the same subspaces as the projection operators of R^+T.

Proof. Similar to Appendix “Task motion—force duality”.

To summarize, it is of particular interest, whether pathological conditions can be encoded in the choice of , and f_m0. Examples include the investigation of the cause-effect of slack muscle tone (e.g., hypotonia or hypertonia) or the increase in co-contraction causing rigidity. The incorporation of the null space complements the original equations and unveils the structure of dynamic redundancy. Additional steps have to be performed to ensure that Eq (28) provides physiologically correct interpretations.

Muscle space equations of motion

In the previous section (“Relation between muscle and joint space quantities”), the various relations between motion and force quantities of the dynamically redundant system were established. In this section we will derive the EoMs that relate the muscle lengthening/shortening accelerations to muscle space forces.

As was previously shown (Eq (28)), the muscle forces f_m can be expressed as the sum of two mutually orthogonal subspaces that span (). We can project Eq (16) onto the muscle space by multiplying both sides from the left with −RM⁻¹, also taking Eqs (25) and (27) into account (29) where represents the muscle space inertia mass matrix, the muscle bias term and is used to project joint space quantities onto the muscle space. Note that the null space term cancel out when multiplied by (Proposition 1). When Eq (29) is solved in an Inverse Dynamics (ID) manner, the quantities of interest are the muscle forces f_m, while , f_m0 are free variables which are selected to satisfy additional modeling criteria.

Proposition 1. [Muscle space inertia mass matrix]

Given that T_R = RR⁺ and the following assertions hold:

1. Λ_m is symmetric () iff M is symmetric.
2. Λ_m remains invariant under the projection operator T_R.
3. .
4. .

Proof. Appendix “Muscle space inertia mass matrix”.

The introduction of the null space term () complements the relationship between muscle and joint space forces (Eq (28)) and the muscle space EoMs (Eq (29)). Without this term, the relations are ill conditioned due to physiological restrictions of the muscles (0 ≼ f_m ≼ f_max). Although a particular solution f_m or for some arbitrary or , respectively does not necessarily satisfy these restrictions, a suitable f_m0 can be selected. Furthermore, several aspects and patterns of muscle co-contractions can be modeled by appropriately selecting f_m0.

Exploitation of kinematic and dynamic redundancy

In the previous section, we derived the EoMs in muscle space. Furthermore, we decomposed the muscle forces (f_m) into two mutually orthogonal subspaces that span the entire muscle space (). Here we will define an effective approach for taking advantage of the null space contribution in order to satisfy the physiological restrictions of the muscles.

Muscle redundancy manifests itself during ID, whereas the null space term does not contribute to the movement in a Forward Dynamics (FD) setting, because . For a given action a particular solution of muscle forces () can be found by initially ignoring the null space contribution (), which is not guaranteed to be physiologically correct (namely ). However, the null space can provide a suitable correction in order to satisfy the physiological constraints. A solution of muscle forces in the task, joint or muscle space respectively can be obtained as follows (note that the MPP solves the least squares problem Definition 2) (30)

It is well-known that muscle dynamics are described by a complex nonlinear system [27–29] that depends on a multitude of factors, such as the strength of the muscles, the muscle activation dynamics, the pennation angle, the elastic properties of the muscle fibers and tendon, the force-length and the force-velocity characteristics, etc. Consequently, these factors will also affect the null space solution component f_m0. Assuming the simplest possible muscle model (31) where represents a vector of muscle activations, a vector specifying the maximum muscle forces and ∘ the Hadamard (elementwise) product. We can impose bounds on the possible solutions of f_m0 in the form of linear inequalities, by noting that Eqs (28) and (31) must be equal (32)

Given an appropriately selected cost function, a unique solution of the linear inequalities can be obtained by solving the constrained optimization problem. A generalization for a nonlinear muscle model [27] is presented below (33) where are the force-length, force-velocity and passive muscle forces.

It is of great interest to understand the structural dependencies of these inequalities in order to outline the feasible muscle force space. For a muscle model and an arbitrary action , the problem is expressed in its most general form (generalization of Eq (33)): (34) where the solutions (f_m0) lie in a convex subspace of (r ≤ m) and the feasible space is bounded (Proposition 2). Note that the action () implicitly encodes the kinematic redundancy (Eq (30)). Finally, the feasible muscle force set is computed as the sum of the particular solution and the null space forces that satisfy the inequalities (35)

Proposition 2. [Feasible null space forces]

The following properties hold true:

1. Eq (34) defines a convex set .
2. Eq (34) is bounded if the inequality is feasible and if Z is full column rank.

Proof. Appendix “Feasible muscle force space”.

Traditional solutions of the muscle redundancy problem [1] (e.g., effort minimization) ignore the possibilities of the system and prove to be inadequate in quantifying the decisive factors, whereas Eqs (34) and (35) have an elegant form that outlines the entire set of possible solutions. More importantly, each solution can be characterized (e.g., minimum effort, high stiffness, rigidity, etc.), enabling classification of the different manifolds (strategies) available to the CNS.

In order to compute the feasible muscle force set , the physiological null space muscle forces f_m0 must be obtained by sampling the space defined by a set of linear inequalities (Eq (34)). The linear inequalities define a polytope (a convex polyhedron) as an intersection of a finite number of half-spaces (hyperplane- or H-representation). According to Proposition 2, this polytope has a finite number of extreme points (vertex- or V-representation), which we use to sample this polytope sufficiently. The conversion from H-representation to V-representation is called vertex enumeration and can be achieved by using either a deterministic [30] or randomized [31] approach. From an arrangement of n hyperplanes in , v vertices are determined in O(n² dv) time. In our problem n = 2 m, d ≤ m (m being number of muscles), thus the time complexity is O(vm³). In this work, we used the lrs library [30], which provides a self-contained ANSI C implementation of the reverse search algorithm for vertex enumeration. After obtaining the extreme points of the polytope, additional solutions are generated by interpolating () between vertices. This process generates samples spanning the entire polytope due to its convexity, as proved earlier. An example of Algorithm 1 is presented (Fig 2) for the following system (36)

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. Sampling of bounded inequality.

The inequality Eq (36) is sampled using Algorithm 1 for different levels of iteration depth (d).

https://doi.org/10.1371/journal.pone.0209171.g002

Algorithm 1 Iterative sampling of a convex bounded polyhedron.

Input: H-representation Ax ≤ b, d: iteration depth

Output: A representative set

1: Find the V-representation using lrs (or some other method)

2: for i = 0 to d do

3:  

4:  for all distinct do

5:   Add to {λ = 0.5}

6:  end for

7: end for

8: return

Muscle space projection and reflexes

Muscle space projection exhibits several advantages in problems related to the segmental level coordination, mainly because of the different muscle related variables (e.g., length, velocity, muscle stiffness) that are directly/indirectly controlled by the CNS [11, 12]. For example, the proprioceptive receptors, located in the spinal cord, constantly measure the evolution of these variables and issue corrective actions [34]. More specifically, the muscle spindle organs [35] measure changes in the muscle length and the Golgi tendon organs [36] measure the force exerted by their respective muscles. This indicates that the aforementioned variables are used in an internal feedback regulation process [3].

We continue with the design of a posture controller that coarsely encapsulates the characteristics of the internal regulation process performed in the spinal cord and the evaluation of system’s response to external disturbances. As the muscle length and its derivative are the regulating variables, muscle space projection is a natural representation in this setting. The following control scheme is adopted for this experiment (37) where represents the muscle length acceleration goals, the desired muscle length positions, l_m(t − τ_so) and the perceived, delayed (τ_so = 20 ms originating from muscle spindle organs [3]) muscle length positions and velocities and k_p, k_d the reflex gains. Note that this control law restores the system to its original posture l_m(t = 0). A force disturbance impulse in an arbitrary direction is applied on the end effector body. The impulse is modeled by a Gaussian function (38) where a controls the magnitude, t₀ the application time and σ the smoothness of the impulse. For the following experiments we will use a = 15, t₀ = 0.1, σ = 0.01 and the perturbation will act in the −x direction. The muscle space EoMs Eq (29) can serve as an ID model-based controller for calculating the required muscle forces f_m (Algorithm 2), which may then be fed to the FD module. The in Algorithm 2 can be arbitrarily selected by the modeler to encode various strategies available by the CNS. In our experiments, is set zero to encode an unbiased strategy, for the sake of simplicity.

Algorithm 2 Muscle space force computation for a given goal using muscle space projection.

Input: muscle length accelerations , null space muscle length velocities

Output: muscle forces f_m

1: Solve Eq (29) for by initially ignoring the null space term

2: Minimize ||f_m0||² for f_m0 subject to Eq (34)

3: Obtain the required muscle forces as

4: return f_m

Fig 4 depicts collectively the simulation results when the system is perturbed and the two reflex loops are active (k_p = 10, k_d = 10 and τ_so = 20 ms). The impact is absorbed and the system returns to its initial configurations (stable response). Conversely, in Fig 5 the “descending” command is disabled (k_p = 0), leaving the response of the reflex loop alone, in which the system reaches a new, stable equilibrium. The muscle space projection suits this kind of studies well as it offers a direct interface with segmental level models. On the contrary, additional transformation of the control variables is required if the problem were expressed in joint space. Different combinations of reflex loop gains, delays and external factors can be evaluated quantitatively, demonstrating the usefulness of this projection.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. Posture analysis of the system in muscle space, including a “descending” command from the CNS.

The first row of figures presents the simulated joint space coordinates and speeds. The second row depicts the evolution of the muscle space quantities of the controller. The model is perturbed and the response of the system is observed. The system absorbs the impact and returns to its initial configuration. The controller gains are k_p = 10, k_d = 10 and the loop delay τ_so = 20 ms.

https://doi.org/10.1371/journal.pone.0209171.g004

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 5. Posture analysis of the system in muscle space without “descending” command from the CNS.

The first row of figures presents the simulated joint space coordinates and speeds. The second row depicts the evolution of the muscle space quantities of the controller. After the perturbation the system settles to a new, stable equilibrium point. The controller gains are k_p = 0, k_d = 10 and the loop delay τ_so = 20 ms.

https://doi.org/10.1371/journal.pone.0209171.g005

Characterization of the feasible muscle force space

In this experiment, the feasible muscle forces that satisfy the task and the physiological muscle constraints are computed. The analysis is restricted on a particular time instance, where we identify the muscle force bounds and correlations between different muscle pairs. The movement of the hand is planned in task space using Eqs (18) and (19) as an ID model-based controller in combination with a Proportional Derivative (PD) tracking scheme (39) where x_d, and denote the desired position, velocity and acceleration of the end effector and k_p = 50, k_d = 5 the tracking gains. The planning problem can be encoded naturally in task space as compared to joint and muscle space controllers. Note that this analysis is independent of the underlying ID representation (Eq (30)). Furthermore, a linear muscle model (Eq (32)) is assumed.

The desired task goal is derived from a smooth sigmoid function that produces bell-shaped velocity profiles in any direction around the initial position of the end effector (40) where x_t, y_t represent the 2 D components of x_t, a = 0.3, b = 4 and t₀ = 1. Different directions of movement are achieved by transforming the goals with H_z(γ), which defines a rotation around the z-axis of an angle γ. Eqs (34) and (35) enable the exploration of the feasible muscle forces. In order to compute the feasible muscle force set (Eq (35)), Eq (34) must be sampled for f_m0. In this experiment, we compute the vertices of the polytope using Algorithm 1 with d = 0. The process is summarized in the following algorithm (Algorithm 3).

Algorithm 3 Calculation of the feasible muscle forces as affected by the moment arm null space, muscle model and the action.

Input: action , muscle model , null space

Output: feasible muscle forces

1: Find the particular muscle force solution (Eq (30))

2: Sample Eq (34) for f_m0 (Algorithm 1)

3: Calculate the feasible muscle force set from Eq (35)

4: return

Fig 6 presents the simulation results for the simplified hand movement (movement along the −x direction, γ = π) collectively. We can inspect the evolution of the generalized coordinates and speeds as well as simulated/desired task goals and the corresponding task forces computed by the controller. Two instances of the simulation are isolated in Fig 7, where the feasible muscle forces that satisfy the task are further analyzed. The resulting feasible spaces are high-dimensional and various useful conclusions can be drawn. The feasible muscle force space changes as the movement progresses, since it depends on the action, the muscle model and the muscle routing properties.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 6. Collective simulation results for the simplified hand movement.

The evolution of the general coordinates and speeds as well as the simulated movement are presented in the figures of the first row. The second row shows the generalized forces, the simulated and desired task goals and the task space forces computed by the controller. The parameters used in this experiment are k_p = 50, k_d = 5, a = 0.3, b = 4, t₀ = 1 and γ = π.

https://doi.org/10.1371/journal.pone.0209171.g006

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 7. The evolution of the feasible force set.

Two instances of the simulated movement along the −x direction are shown to demonstrate the evolution of the feasible muscle space. The box plot depicts the force variability of each muscle as contributed to the redundant nature of the system. The correlation matrix shows the positive or negative correlation between the different muscle pairs.

https://doi.org/10.1371/journal.pone.0209171.g007

The box plot shows the force variability of each muscle, which is attributed to the redundant nature of the system. From a control point of view, the variability of each muscle seems to be inversely proportional to the importance of its contribution to the current movement, following the Uncontrolled Manifold Hypothesis (UMH) [9]. Fig 7 presents two instances of the simulated movement, one at the onset of the movement (0.5 s) and one at the peak of the required effort (0.9 s), permitting a qualitative comparison the feasible muscle forces. As the force requirements are small in the beginning of the movement (Fig 7a), the feasible muscle forces span the entire space. On the contrary (Fig 7b), as the demand for movement increases the agonists retain higher force profiles, while the antagonists’ contribution is attenuated. The provided correlation matrix between different muscle groups reveals agonist-antagonist relations, which vary as the movement progresses. It can be concluded that there is a significant systematic correlation between different muscle pairs that emerges from the muscle routing properties and the functional constraints of the task.

The effect of null space on the joint reaction loads

The importance of evaluating the feasible muscle forces is demonstrated in the context of joint reaction analysis. An accurate estimation of the muscle forces is essential for the assessment of joint reaction loads. Consequently, the null space contributions can significantly alter the reaction forces without affecting the movement.

A benchmark gait movement, available from the OpenSim dataset [33] was used for this analysis. In a typical experimental setup the motion and externally applied forces are recorded. Given these recordings, Inverse Kinematics (IK) and ID are performed in order to assess the model kinematics and kinetics required to track the experimental measurements [37]. Instead of estimating the muscle forces using Static Optimization (SO) or some other method [1, 38], we can recall Eq (28) and solve for f_m accounting for the null space muscle forces, which are present only when projecting from low- to high-dimensional space. Different realizations of muscle forces were calculated by appropriately sampling the feasible muscle force space (Eqs (34) and (35)). Finally, multiple joint reaction analyses were performed to evaluate the effect of the feasible muscle forces on the reaction loads.

Fig 8 presents the normalized (with respect to body weight) reaction forces on the hip joint during walking with the heel strike and toe-off events annotated accordingly. We observe that the results obtained from OpenSim joint reaction analysis predict low reaction load levels, since the minimum muscle effort criterion is used to compute the muscle forces, ignoring muscle co-contraction. On the contrary, it is possible to calculate the feasible reactions without making any prior assumption, which can limit the scope and extent of the analysis. Last and perhaps most importantly, the large range of possible values confirms that misinterpretation of the results is possible if the null space solutions are ignored.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 8. The effect of the null space muscle forces on the joint reaction loads.

The normalized (with respect to body weight) reaction loads on the hip joint during walking are reported along with the heel strike and toe-off events. The shaded area is the reaction force range as attributed to the null space solutions of muscle forces. The red dotted line are the results obtained from OpenSim. The large range in the reaction loads confirms that misinterpretation of the results is possible if the null space solutions are ignored.

https://doi.org/10.1371/journal.pone.0209171.g008

Projection operators

Let T_A = AA⁺ and F_A = A⁺ A be two operators derived from matrix A, where A⁺ is the MPP matrix. The following proofs rely on the Singular Value Decomposition (SVD) theorem (A = UΣV^T, A⁺ = VΣ⁺U^T) and the properties of the MPP (Definition 2).

1. (a). and . where (AA⁺)^T = AA⁺ and A⁺ AA⁺ = A⁺ are properties of the MPP (Definition 2).
2. (b). and . Similar to (a) with (A⁺ A)^T = A⁺ A and AA⁺ A = A (Definition 2).
3. (c). T_A A = AF_A = A and A⁺ T_A = F_A A⁺ = A⁺ can be proven using AA⁺ A = A and A⁺ AA⁺ = A⁺.
4. (d). T_A is the projector onto the . where V^T V = I and r is the rank and n the nullity of A.
5. (e). F_A is the projector onto the . where U^T U = I.
6. (f). projects onto the , because and are mutually orthogonal complement (Eq (6)).
7. (g). projects onto the , because and are mutually orthogonal complement (Eq (6)).
8. (h). and .

Task motion—Force duality

Given that is the MPP matrix and the definition of SVD (), the following hold true:

1. (a).
2. (b).

thus their projection operators span the same subspaces.

Muscle space inertia mass matrix

1. (a). is symmetric () iff M is symmetric.
2. (b). remains invariant under the projection operator T_R.
3. (c). .
4. (d). . This can be shown using (b).

Feasible muscle force space

1. (a). Let’s prove that Eq (34) defines a convex set . For any and λ ∈ [0, 1], let then which proves the convexity property.
2. (b). Let’s prove that Eq (34) is bounded when the inequality is feasible and Z has full column rank. From the definition, this inequality has always the following form (41) where a an b are finite vectors in . If the system of inequalities is feasible and has full column rank, then f_m0 is upper and lower bounded. Even if the problem is feasible, if is rank deficient, then , thus the feasible space is not bounded. Note, however, that we can always reduce the null space projection to full column rank by eliminating linearly dependent columns in .