Fig 1.
We explore the interactions of muscle synergies with endpoint stiffness synthesis and energy consumption in the context of feasible activations to meet task constraints.
Fig 2.
We use a 6-muscle planar arm model to quantify the effects of synergies on endpoint stiffness and energy consumption within the workspace. For any posture in the workspace of the arm, neural commands to the muscles can set the active endpoint stiffness of the limb—visualized as an ellipse of a particular size, shape (i.e., eccentricity), and orientation.
Fig 3.
Reformulation of stiffness equations.
Obtaining Eq 10 requires that we transform the matrices in Eq 7. In this way, the vector can be expressed as a set of linear equations in
. These linear equations become the linear constraints for endpoint stiffness that we use to solve the quadratic programming problem to minimize energy. (•) denotes element-by-element multiplication, and Ri is the ith row of R.
Fig 4.
We simulate experimentally-observed synergies that group the mono-articular shoulder muscles and all the muscles crossing the elbow. Briefly, a synergy is the correlated activity of muscle activations. Each synergy is independently controlled and synergistically drives its muscles according to specific muscle weighting parameters. In this case, we simulate how two synergies drive the six muscles of an arm as per the number of synergies and weights reported by [19].
Fig 5.
Visualization of mechanical and energetic constraints.
A. Consider a limb with one rotational joint driven by 3 muscles. We show the parameters and variables needed to implement Eq 14. B. As described elsewhere in detail [34–36, 41, 42], the set of all physiologically feasible neural commands to a limb that can be visualized as a positive hypercube of as many dimensions as there are independently controllable muscles (3 in this case). Each constraint that defines the mechanical task reduces the set of feasible neural commands to a subset of that hypercube. Therefore, the feasible activation set for a task is the intersection of those constraints that lie within the hypercube. If the constraints are linear functions of activation, they reduce the subset of feasible neural commands to a hyperplane. We see this for the linear constraints defining the size, shape (i.e., eccentricity) and orientation of the desired endpoint stiffness ellipse (blue plane); and enforcing static equilibrium with zero endpoint force (green plane). Their intersections with the hypercube are shown schematically as triangular polygons. The feasible activation set that meets both constraints, if it exists, is the black line—which naturally contains infinitely many points (i.e., muscle redundancy). The individual solution with the minimal energetic cost is given by the point where the line is tangential to the smallest spherical manifold defined by the quadratic cost function in Eq 11. C. A muscle synergy can similarly be visualized as a constraint that ties the activations of several muscles in an obligatory way, which in this 3-dimensional example reduces the set of all physiologically feasible neural commands from a cube to a plane. This reduction in independently controllable muscle actions has the inevitable consequence of reducing—or even annihilating—the ability to meet multiple endpoint stiffness and energetic constrains as shown in our results.
Fig 6.
Method to find realizable endpoint stiffnesses throughout the workspace.
Using our 6-muscle planar arm model, we are able to iteratively check for realizable endpoint stiffness ellipses throughout the workspace. In this case, the ability to set its orientation in 5 degree increments for a given stiffness ellipse shape (i.e., eccentricity). As the posture of the limb changes to make the endpoint visit each point in the workspace, the associated changes in its Jacobian matrix and the constraints of the task interact to affect the realizable endpoint stiffness ellipses. Red regions indicate the locations of the endpoint where all orientations are realizable (i.e., 100% or a fraction of 1), whereas deep blue and black regions indicate the locations in the workspace for which there is very limited or non-existent ability to arbitrarily control the orientation of the endpoint stiffness ellipse (i.e., 20%—0% or fractions of 0.2—0).
Fig 7.
The fraction of realizable stiffness orientations is heavily influenced by arm posture and stiffness ellipse shape (i.e., eccentricity measured by condition number, or ratio of length of the major to the minor axis).
Fig 8.
Synergies reduce the ability to control endpoint stiffness.
Shoulder and elbow synergies (Fig 4) cause covariation of stiffness ellipse eccentricity with orientation and limit the range of ellipse orientations. Note that as the elbow to shoulder activation ratio changes from 10−1 to 10 the shape and orientation (direction of major axis) of the ellipse follow an obligatory relationship. This is expected from the fact that adding a synergy—correlated activation of multiple muscles—reduces the set of feasible activations, Fig 5.
Fig 9.
Some arm postures compromise the ability to minimize energy.
Black regions indicate locations in the workspace (i.e, limb postures) where meeting the desired endpoint stiffness leaves no room to minimize energy. These regions are also heavily influenced by the desired stiffness ellipse shape (i.e., eccentricity). Even when minimizing energy is possible, that ability seldom reaches a reduction of 50% and is typically < 10–30%.
Fig 10.
Summary of results and comparison to prior work.
Implementing fewer synergies (i.e., independently controllable groupings of muscle activations) improves the independent control over the size, shape and orientation of the stiffness ellipses, as well as the energy consumption associated with each solution.
Fig 11.
Regardless of their details, synergies bring about functional limitations, Part I.
In addition to the synergies reported by Osu and Gomi (1999), we explored the functional consequences of five additional potential synergies (i.e., weights in the correlations among muscle activations). The first two are shown here as Cases 2 and 3. The remaining three are shown in Fig 12.
Fig 12.
Regardless of their details, synergies bring about functional limitations, Part II.
Functional consequences of three additional potential synergies, for total of five beyond those reported by Osu and Gomi (1999), as mentioned in Fig 11. Because synergies invariably imply a loss of control degrees of freedom (i.e., fewer independently controllable muscles as seen in Fig 5), it is to be expected that they all reduce the controllability of the size, shape and orientation of the stiffness ellipses as shown here and in Fig 11. Therefore, our results are generalizable to the concept of synergies in general, and are not limited to the particulars of any one specific synergy.