The organization of intralimb and interlimb synergies in response to different joint dynamics

Abstract

We sought to understand differences in joint coordination between the dominant and nondominant arms when performing repetitive tasks. The uncontrolled manifold approach was used to decompose the variability of joint motions into components that reflect the use of motor redundancy or movement error. First, we hypothesized that coordination of the dominant arm would demonstrate greater use of motor redundancy to compensate for interaction forces than would coordination of the nondominant arm. Secondly, we hypothesized that when interjoint dynamics were more complex, control of the interlimb relationship would remain stable despite differences in control of individual hand paths. Healthy adults performed bimanual tracing of two orientations of ellipses that resulted in different magnitudes of elbow interaction forces. For the dominant arm, joint variance leading to hand path error was the same for both ellipsis orientations, whereas joint variance reflecting the use of motor redundancy increased when interaction moment was highest. For the nondominant arm, more joint error variance was found when interaction moment was highest, whereas motor redundancy did not differ across orientations. There was no apparent difference in interjoint dynamics between the two arms. Thus, greater skill exhibited by the dominant arm may be related to its ability to utilize motor redundancy to compensate for the effect of interaction forces. However, despite the greater error associated with control of the nondominant hand, control of the interlimb relationship remained stable when the interaction moment increased. This suggests separate levels of control for inter- versus intra-limb coordination in this bimanual task.

Appendix

To compute the equations of motion, the Lagrangian of the arm was formulated in terms of joint angles and joint velocities, and then solved analytically using Mathematica^® 5.0 (Wolfram Research, Inc.). The general formulation of equations of motion can be represented as

$$ M(\theta ) \times \ddot{\theta } + H(\theta ,\dot{\theta }) = \tau_{\text{muscle}} $$

(5)

where $H(\theta ,\dot{\theta })$ is a combination of Coriolis and centrifugal forces acting at each joint and its magnitude depends on the motion and position of other joints. τ _muscle is the generalized muscle moment (MM) that includes active muscle activity and passive forces arising from the viscoelastic properties of muscles, tendons, ligaments and periarticular soft tissues. There is no gravitational moment, since movement is constrained in a horizontal plane. M(θ) is a 4 × 4, configuration-dependent inertial matrix; $\ddot{\theta },\;\dot{\theta }\;{\text{and}}\;\theta$ are the vectors of joint acceleration, velocity and angle, respectively. The inertial matrix M(θ) has diagonal entries corresponding to the inertia of a given segment of interest and off-diagonal entries capturing the effect on each joint based on the acceleration of the other joints. By separating out the two components of the inertial matrix, M _d(θ) and M _nd(θ), Eq. 5 can be written as:

$$ M_{\text{d}} (\theta ) \times \ddot{\theta } = - M_{\text{nd}} (\theta ) \times \ddot{\theta } - H(\theta ,\dot{\theta }) + \tau_{\text{muscle}} $$

(6)

Equation 6 can be further represented as the following, by grouping the appropriate terms for interaction moments

$$ {\text{NM}} = {\text{IM}} + {\text{MM}} $$

(7)

The net moment (NM) is proportional to joint acceleration and is directly responsible for motion of this joint. IM is the interaction moment that depends on mutual interactions with the other joints. MM represents the generalized muscle moment.

The individual terms of the M and H matrices are listed below. Note that the diagonal terms are M _1,1, M_2,2, M_3,3, M_4,4, while the off diagonal terms are M _i,j, where i ≠ j; i and j = 1, 2, 3, 4. The same arrangement applies to the H matrix. Notation: θ _i angle of the ith joint; l _i length of the ith segment; r _i position of the center of mass of the ith joint from the proximal end of that segment; m _i mass of the ith segment; I _i moment of inertia of the ith segment. The number representing each arm segment and joint angle (in parentheses): 1 clavicle (scapula); 2 upper arm (shoulder); 3 forearm (elbow); 4 hand (wrist).

$$ \begin{aligned} M(\theta )_{1,1} & = I_{1} + I_{2} + I_{3} + I_{4} + m_{1} \cdot r_{1}^{2} + (m_{2} + m_{3} + m_{4} ) \cdot l_{1}^{2} + (m_{3} + m_{4} ) \cdot l_{2}^{2} + m_{2} \cdot r_{2}^{2} + m_{4} \cdot \\ & l_{3}^{2} + m_{3} \cdot r_{3}^{2} + m_{4} \cdot r_{4}^{2} + m_{3} \cdot (2 \cdot l_{1} \cdot l_{2} \cdot \hbox{cos}(\theta_{2} ) + 2 \cdot l_{1} \cdot r_{3} \cdot \hbox{cos}(\theta_{2} + \theta_{3} ) + 2 \cdot l_{2} \cdot \\ & r_{3} \cdot \hbox{cos}(\theta_{3} )) + 2 \cdot m_{2} \cdot l_{1} \cdot r_{2} \hbox{cos}(\theta_{2} ) + m_{4} \cdot (2 \cdot l_{1} \cdot l_{2} \cdot \hbox{cos}(\theta_{2} ) + 2 \cdot l_{1} \cdot l_{3} \cdot \\ & \hbox{cos}(\theta_{2} + \theta_{3} ) + 2 \cdot l_{1} \cdot r_{4} \cdot \hbox{cos}(\theta_{2} + \theta_{3} + \theta_{4} ) + 2 \cdot l_{2} \cdot l_{3} \cdot \hbox{cos}(\theta_{3} ) + 2 \cdot l_{2} \cdot r_{4} \cdot \\ & \hbox{cos}(\theta_{3} + \theta_{4} ) + 2 \cdot l_{3} \cdot r_{4} \cdot \hbox{cos}(\theta_{4} )) \\ \end{aligned} $$

$$ \begin{aligned} M(\theta )_{1,2} & = I_{2} + I_{3} + I_{4} + m_{2} \cdot r_{2}^{2} + m_{3} \cdot r_{3}^{2} + m_{4} \cdot r_{4}^{2} + m_{2} \cdot l_{1} \cdot r_{2} \cdot \hbox{cos}(\theta_{2} ) + (m_{3} + m_{4} ) \cdot l_{2}^{2} + \\ & m_{4} \cdot l_{3}^{2} + m_{3} \cdot (l_{1} \cdot l_{2} \cdot \hbox{cos}(\theta_{2} ) + l_{1} \cdot r_{3} \cdot \hbox{cos}(\theta_{2} + \theta_{3} ) + 2 \cdot l_{2} \cdot r_{3} \cdot \hbox{cos}(\theta_{3} )) + m_{4} \cdot \\ & (l_{1} \cdot l_{2} \cdot \hbox{cos}(\theta_{2} ) + l_{1} \cdot l_{3} \cdot \hbox{cos}(\theta_{2} + \theta_{3} ) + l_{1} \cdot r_{4} \cdot \hbox{cos}(\theta_{2} + \theta_{3} + \theta_{4} ) + 2 \cdot l_{2} \cdot l_{3} \cdot \\ & \hbox{cos}(\theta_{3} ) + 2 \cdot l_{2} \cdot r_{4} \cdot \hbox{cos}(\theta_{3} + \theta_{4} ) + 2 \cdot l_{3} \cdot r_{4} \cdot \hbox{cos}(\theta_{4} )) \\ \end{aligned} $$

$$ \begin{aligned} M(\theta )_{1,3} & = I_{3} + I_{4} + m_{3} \cdot r_{3}^{2} + m_{4} \cdot r_{4}^{2} + m_{4} \cdot l_{3}^{2} + m_{3} \cdot (l_{1} \cdot r_{3} \cdot \hbox{cos}(\theta_{2} + \theta_{3} ) + l_{2} \cdot r_{3} \cdot \hbox{cos}(\theta_{3} )) + \\ & m_{4} \cdot (l_{1} \cdot l_{3} \cdot \hbox{cos}(\theta_{2} + \theta_{3} ) + l_{1} \cdot r_{4} \cdot \hbox{cos}(\theta_{2} + \theta_{3} + \theta_{4} ) + l_{2} \cdot l_{3} \cdot \hbox{cos}(\theta_{3} ) + l_{2} \cdot \\ & r_{4} \cdot \hbox{cos}(\theta_{3} + \theta_{4} ) + 2 \cdot l_{3} \cdot r_{4} \cdot \hbox{cos}(\theta_{4} )) \\ M(\theta )_{1,4} & = I_{4} + m_{4} \cdot r_{4}^{2} + m_{4} \cdot (l_{1} \cdot r_{4} \cdot \hbox{cos}(\theta_{2} + \theta_{3} + \theta_{4} ) + l_{2} \cdot r_{4} \cdot \hbox{cos}(\theta_{3} + \theta_{4} ) + l_{3} \cdot r_{4} \cdot \hbox{cos}(\theta_{4} )) \\ M(\theta )_{2,1} & = M(\theta )_{1,2} \\ M(\theta )_{2,2} & = I_{2} + I_{3} + I_{4} + m_{2} \cdot r_{2}^{2} + m_{3} \cdot l_{2}^{2} + m_{3} \cdot r_{3}^{2} + 2 \cdot m_{3} \cdot l_{2} \cdot r_{3} \cdot \hbox{cos}(\theta_{3} ) + \\ & m_{4} \cdot (l_{2}^{2} + l_{3}^{2} + r_{4}^{2} + 2 \cdot l_{2} \cdot l_{3} \cdot \hbox{cos}(\theta_{3} )) + 2 \cdot l_{2} \cdot r_{4} \cdot \hbox{cos}(\theta_{3} + \theta_{4} ) + 2 \cdot l_{3} \cdot r_{4} \cdot \hbox{cos}(\theta_{4} )) \\ \end{aligned} $$

$$ \begin{aligned} M(\theta )_{2,3} & = I_{4} + I_{3} + m_{3} \cdot r_{3}^{2} + m_{3} \cdot l_{2} \cdot r_{3} \cdot \hbox{cos}(\theta_{3} ) + m_{4} \cdot (l_{3}^{2} + r_{4}^{2} + l_{2} \cdot l_{3} \cdot \hbox{cos}(\theta_{3} ) + \\ & l_{2} \cdot r_{4} \cdot \hbox{cos}(\theta_{3} + \theta_{4} ) + 2 \cdot l_{3} \cdot r_{4} \cdot \hbox{cos}(\theta_{4} )) \\ M(\theta )_{2,4} & = I_{4} + m_{4} \cdot r_{4}^{2} + m_{4} \cdot l_{2} \cdot r_{4} \cdot \hbox{cos}(\theta_{3} + \theta_{4} ) + m_{4} \cdot l_{3} \cdot r_{4} \cdot \hbox{cos}(\theta_{4} ) \\ M(\theta )_{3,1} & = M(\theta )_{1,3} \\ M(\theta )_{3,2} & = M(\theta )_{2,3} \\ M(\theta )_{3,3} & = I_{3} + I_{4} + m_{3} \cdot r_{3}^{2} + m_{4} \cdot (l_{3}^{2} + r_{4}^{2} + 2 \cdot l_{3} \cdot r_{4} \cdot \hbox{cos}(\theta_{4} )) \\ M(\theta )_{3,4} & = I_{4} + m_{4} \cdot r_{4}^{2} + m_{4} \cdot l_{3} \cdot r_{4} \cdot \hbox{cos}(\theta_{4} ) \\ M(\theta )_{4,1} & = M(\theta )_{1,4} \\ M(\theta )_{4,2} & = M(\theta )_{2,4} \\ M(\theta )_{4,3} & = M(\theta )_{3,4} \\ M(\theta )_{4,4} & = m_{4} \cdot r_{4}^{2} + I_{4} \\ \end{aligned} $$

$$ \begin{aligned} H(\theta ,\dot{\theta })_{1} & = (2 \cdot \mathop {\dot{\theta }}\nolimits_{1} \cdot \mathop {\dot{\theta }}\nolimits_{3} + 2 \cdot \mathop {\dot{\theta }}\nolimits_{2} \cdot \mathop {\dot{\theta }}\nolimits_{3} + \mathop {\dot{\theta }}\nolimits_{3}^{2} ) \cdot ( - m_{3} \cdot l_{1} \cdot r_{3} \cdot \hbox{sin}(\theta_{2} + \theta_{3} ) - m_{3} \cdot l_{2} \cdot r_{3} \\ & \hbox{sin}(\theta_{3} )) + (\mathop {\dot{\theta }}\nolimits_{2}^{2} + 2 \cdot \mathop {\dot{\theta }}\nolimits_{1} \cdot \mathop {\dot{\theta }}\nolimits_{2} ) \cdot (\hbox{sin}(\theta_{2} ) \cdot ( - m_{2} \cdot l_{1} \cdot r_{2} - m_{3} \cdot l_{1} \cdot l_{2} ) \\ & - m_{3} \cdot l_{1} \cdot r_{3} \cdot \hbox{sin}(\theta_{2} + \theta_{3} )) - m_{4} \cdot ((2 \cdot \mathop {\dot{\theta }}\nolimits_{1} \cdot \mathop {\dot{\theta }}\nolimits_{4} + 2 \cdot \mathop {\dot{\theta }}\nolimits_{2} \cdot \mathop {\dot{\theta }}\nolimits_{4} + 2 \cdot \mathop {\dot{\theta }}\nolimits_{3} \cdot \mathop {\dot{\theta }}\nolimits_{4} + \mathop {\dot{\theta }}\nolimits_{4}^{2} ) \\ & \cdot (l_{3} \cdot r_{4} \cdot \hbox{sin}(\theta_{4} ) + l_{2} \cdot r_{4} \cdot \hbox{sin}(\theta_{3} + \theta_{4} ) + l_{1} \cdot r_{4} \cdot \hbox{sin}(\theta_{2} + \theta_{3} + \theta_{4} )) + (2 \cdot \mathop {\dot{\theta }}\nolimits_{1} \cdot \mathop {\dot{\theta }}\nolimits_{3} + \\ & 2 \cdot \mathop {\dot{\theta }}\nolimits_{2} \cdot \mathop {\dot{\theta }}\nolimits_{3} + \mathop {\dot{\theta }}\nolimits_{3}^{2} ) \cdot (l_{2} \cdot r_{4} \cdot \hbox{sin}(\theta_{3} + \theta_{4} ) + l_{1} \cdot l_{3} \cdot \hbox{sin}(\theta_{2} + \theta_{3} ) + l_{2} \cdot l_{3} \cdot \hbox{sin}(\theta_{3} ) + \\ & l_{1} \cdot r_{4} \cdot \hbox{sin}(\theta_{2} + \theta_{3} + \theta_{4} )) + (2 \cdot \mathop {\dot{\theta }}\nolimits_{1} \cdot \mathop {\dot{\theta }}\nolimits_{2} + \mathop {\dot{\theta }}\nolimits_{2}^{2} ) \cdot ({\l}_{1} \cdot r_{4} \cdot \hbox{sin}(\theta_{2} + \theta_{3} + \theta_{4} ) + l_{1} \cdot l_{3} \cdot \\ & \hbox{sin}(\theta_{2} + \theta_{3} ) + l_{1} \cdot l_{2} \cdot \hbox{sin}(\theta_{2} )) \\ \end{aligned} $$

$$ \begin{aligned} H(\theta ,\dot{\theta })_{2} & = \mathop {\dot{\theta }}\nolimits_{1}^{2} \cdot \hbox{sin}(\theta_{2} ) \cdot (m_{2} \cdot l_{1} \cdot r_{2} + m_{3} \cdot l_{1} \cdot l_{2} ) + m_{3} \cdot r_{3} \cdot l_{1} \cdot \hbox{sin}(\theta_{2} + \theta_{3} ) \cdot \mathop {\dot{\theta }}\nolimits_{1}^{2} - \\ & m_{3} \cdot l_{2} \cdot r_{3} \cdot \hbox{sin}(\theta_{3} ) \cdot (2 \cdot \mathop {\dot{\theta }}\nolimits_{3} \cdot \mathop {\dot{\theta }}\nolimits_{1} + 2 \cdot \mathop {\dot{\theta }}\nolimits_{3} \cdot \mathop {\dot{\theta }}\nolimits_{2} + \mathop {\dot{\theta }}\nolimits_{3}^{2} ) + \mathop {\dot{\theta }}\nolimits_{1}^{2} \cdot (m_{4} \cdot l_{1} \cdot l_{2} \cdot \hbox{sin}(\theta_{2} ) + \\ & m_{4} \cdot l_{1} \cdot l_{3} \cdot \hbox{sin}(\theta_{2} + \theta_{3} ) + m_{4} \cdot l_{1} \cdot r_{4} \cdot \hbox{sin}(\theta_{2} + \theta_{3} + \theta_{4} )) - m_{4} \cdot ((l_{2} \cdot l_{3} \cdot \hbox{sin}(\theta_{3} ) \\ & + l_{2} \cdot r_{4} \cdot \hbox{sin}(\theta_{3} + \theta_{4} )) \cdot (2 \cdot \mathop {\dot{\theta }}\nolimits_{3} \cdot \mathop {\dot{\theta }}\nolimits_{1} + 2 \cdot \mathop {\dot{\theta }}\nolimits_{3} \cdot \mathop {\dot{\theta }}\nolimits_{2} + \mathop {\dot{\theta }}\nolimits_{3}^{2} ) + (l_{2} \cdot r_{4} \cdot \hbox{sin}(\theta_{3} + \theta_{4} ) + \\ & l_{3} \cdot r_{4} \cdot \hbox{sin}(\theta_{4} )) \cdot (2 \cdot \mathop {\dot{\theta }}\nolimits_{1} \cdot \mathop {\dot{\theta }}\nolimits_{4} + 2 \cdot \mathop {\dot{\theta }}\nolimits_{2} \cdot \mathop {\dot{\theta }}\nolimits_{4} + 2 \cdot \mathop {\dot{\theta }}\nolimits_{3} \cdot \mathop {\dot{\theta }}\nolimits_{4} + \mathop {\dot{\theta }}\nolimits_{4}^{2} ) \\ \end{aligned} $$

$$ \begin{aligned} H(\theta ,\dot{\theta })_{3} & = (\mathop {\dot{\theta }}\nolimits_{1}^{2} + \mathop {\dot{\theta }}\nolimits_{2}^{2} + 2 \cdot \mathop {\dot{\theta }}\nolimits_{1} \cdot \mathop {\dot{\theta }}\nolimits_{2} ) \cdot (m_{3} \cdot l_{2} \cdot r_{3} \cdot \hbox{sin}(\theta_{3} ) + m_{4} \cdot l_{2} \cdot l_{3} \cdot \hbox{sin}(\theta_{3} ) + m_{4} \cdot l_{2} \cdot r_{4} \cdot \\ & \hbox{sin}(\theta_{3} + \theta_{4} )) + \mathop {\dot{\theta }}\nolimits_{1}^{2} \cdot (m_{3} \cdot l_{1} \cdot r_{3} \cdot \hbox{sin}(\theta_{2} + \theta_{3} ) + m_{4} \cdot{\l}_{1} \cdot l_{3} \cdot \hbox{sin}(\theta_{2} + \theta_{3} ) \\ & + m_{4} \cdot l_{1} \cdot r_{4} \cdot \hbox{sin}(\theta_{2} + \theta_{3} + \theta_{4} )) - m_{4} \cdot l_{3} \cdot r_{4} \cdot \hbox{sin}(\theta_{4} ) \cdot (2 \cdot \mathop {\dot{\theta }}\nolimits_{4} \cdot \mathop {\dot{\theta }}\nolimits_{1} + 2 \cdot \mathop {\dot{\theta }}\nolimits_{4} \cdot \mathop {\dot{\theta }}\nolimits_{2} + \\ & 2 \cdot \mathop {\dot{\theta }}\nolimits_{4} \cdot \mathop {\dot{\theta }}\nolimits_{3} + \mathop {\dot{\theta }}\nolimits_{4}^{2} ) \\ \end{aligned} $$

$$ \begin{aligned} H(\theta ,\dot{\theta })_{4} & = \mathop {\dot{\theta }}\nolimits_{1}^{2} \cdot m_{4} \cdot l_{1} \cdot r_{4} \cdot \hbox{sin}(\theta_{2} + \theta_{3} + \theta_{4} ) + (\mathop {\dot{\theta }}\nolimits_{1} + \mathop {\dot{\theta }}\nolimits_{2} )^{2} \cdot m_{4} \cdot l_{2} \cdot r_{4} \cdot \hbox{sin}(\theta_{3} + \theta_{4} ) + \\ & m_{4} \cdot l_{3} \cdot r_{4} \cdot \hbox{sin}(\theta_{4} ) \cdot ((\mathop {\dot{\theta }}\nolimits_{1} + \mathop {\dot{\theta }}\nolimits_{2} )^{2} + 2 \cdot \mathop {\dot{\theta }}\nolimits_{1} \cdot \mathop {\dot{\theta }}\nolimits_{3} + 2 \cdot \mathop {\dot{\theta }}\nolimits_{2} \cdot \mathop {\dot{\theta }}\nolimits_{3} + \mathop {\dot{\theta }}\nolimits_{3}^{2} ) \\ \end{aligned} $$