The leading joint hypothesis for spatial reaching arm motions

Abstract

The leading joint hypothesis (LJH), developed for planar arm reaching, proposes that the interaction torques experienced by the proximal joint are low compared to the corresponding muscle torques. The human central nervous system could potentially ignore these interaction torques at the proximal (leading) joint with little effect on the wrist trajectory, simplifying joint-level control. This paper investigates the extension of the LJH to spatial reaching. In spatial motion, a number of terms in the governing equation (Euler’s angular momentum balance) that vanish for planar movements are non-trivial, so their contributions to the joint torque must be classified as net, interaction or muscle torque. This paper applies definitions from the literature to these torque components to establish a general classification for all terms in Euler’s equation. This classification is equally applicable to planar and spatial motion. Additionally, a rationale for excluding gravity torques from the torque analysis is provided. Subjects performed point-to-point reaching movements between targets whose locations ensured that the wrist paths lay in various portions of the arm’s spatial workspace. Movement kinematics were recorded using electromagnetic sensors located on the subject’s arm segments and thorax. The arm was modeled as a three-link kinematic chain with idealized spherical and revolute joints at the shoulder and elbow. Joint torque components were computed using inverse dynamics. Most movements were ‘shoulder-led’ in that the interaction torque impulse was significantly lower than the muscle torque impulse for the shoulder, but not the elbow. For the few elbow-led movements, the interaction impulse at the elbow was low, while that at the shoulder was high, and these typically involved large elbow and small shoulder displacements. These results support the LJH and extend it to spatial reaching motion.

Appendix

The analytical expressions for the net, interaction, and muscle torques for the shoulder and elbow joints developed in Ambike (2011) are provided here. The thorax is assumed to be stationary. The elbow is modeled as a single-DOF revolute joint with the rotation axis along the Z axis of the coordinate frame fixed in the upper arm, as recommended by the ISB protocol (Wua et al. 2005). The shoulder is modeled as a ball-and-socket joint allowing three degrees of rotational freedom for the humerus. The glenohumeral center was assumed as the joint center. The wrist joint is ignored. The hand is incorporated as a point mass at the end of the forearm, as seen in Fig. 8. The mass and center of mass of the combined ‘forearm-hand’ are computed and used in the torque calculations. In Ambike (2011), the development in Feltner and Dapena (1989), Gagnon and Gagnon (1992), Winter (2005) is modified by introducing movement of the forearm relative to the upper arm.

Fig. 8

Free body diagrams of upper arm and forearm. Hand is modeled as point mass rigidly attached to forearm

Figure 8 shows the global coordinate frame, frames attached to the forearm (F) and upper arm (U), and the free body diagrams of the arm segments. All variables are defined in Table 3. For a vector quantity, the leading superscript indicates the coordinate frame in which the vector is expressed, but the superscript is omitted for the global frame. The Newton–Euler approach (Winter 2005) for developing the equations of motions is more suitable than the Lagrangian approach (Greenwood 1988) for implementing the partitioning scheme outlined in "Data recording and analysis". Newton’s second law applied to each segment yields the joint reaction forces, $\overline{R}_E$ and $\overline{R}_F$, at the elbow and the shoulder, respectively. The partitioning scheme is first applied to the reaction forces to obtain reaction components that can be attributed to motions of only the elbow ($\overline{R}_{EF}, \overline{R}_{SF}$), only the shoulder ($\overline{R}_{EU}, \overline{R}_{SU}$), both joints ($\overline{R}_{Eint}, \overline{R}_{Sint}$), and gravity ($\overline{R}_{Eg}, \overline{R}_{Sg}$).

$$ \overline{R}_E = \overline{R}_{EF} + \overline{R}_{EU} + \overline{R}_{Eint} + \overline{R}_{Eg}, $$

where

$$ \begin{aligned} \overline{R}_{EF} &= M_{fh}\overline{a}_{pf}, \quad \overline{R}_{EU} = M_{fh}\overline{a}_{pu}, \\ \overline{R}_{Eint} &= M_{fh}\overline{a}_{pint}, \quad \overline{R}_{Eg} = -M_{fh}\overline{g}. \\ \end{aligned} $$

Table 3 Symbols for the reaction forces and joint torque expressions

Similarly, for the shoulder joint,

$$ \overline{R}_{S} = \overline{R}_{SU} + \overline{R}_{SF} + \overline{R}_{Sint} + \overline{R}_{Sg}, $$

where

$$ \begin{aligned} \overline{R}_{SU} &= m_u\overline{a}_u + \overline{R}_{EU}, \quad \overline{R}_{SF} = \overline{R}_{EF}, \\ \overline{R}_{Sint} &= \overline{R}_{Eint}, \quad \overline{R}_{Sg} = -(M_{fh}+m_u)\overline{g}. \\ \end{aligned} $$

Two applications of Euler’s equation yield the net, interaction, and gravity torques at the elbow,

$$ \begin{aligned} ^F\overline{T}_{Enet} = & I_F \cdot ^F\overline{\alpha}_{fu} + ^F\overline{\omega}_{fu} \times I_F \cdot ^F\overline{\omega}_{fu} \\ &\quad-^F\overline{r}_{E/CF} \times ^F\overline{R}_{EF}, \end{aligned} $$

(5)

$$ \begin{aligned} ^F\overline{T}_{Eint} &=\, I_F\cdot (^F\overline{\alpha}_u + ^F\overline{\omega}_u \times ^F\overline{\omega}_{fu}) \\ &\quad+^F\overline{\omega}_{fu} \times I_F \cdot ^F\overline{\omega}_u \\ &\quad+^F\overline{\omega}_u \times I_F \cdot (^F\overline{\omega}_u +^F\overline{\omega}_{fu}) \\ &\quad- ^F\overline{r}_{E/CF} \times (^F\overline{R}_{EU} + ^F\overline{R}_{Eint}), \end{aligned} $$

(6)

$$ ^F\overline{T}_{Eg} = -^F\overline{r}_{E/CF} \times ^F\overline{R}_{Eg} - ^F\overline{r}_{W/CF} \times m_h ^F\overline{g}, $$

(7)

and at the shoulder,

$$ \begin{aligned} ^U\overline{T}_{Snet} & =\,I_U \cdot ^U\overline{\alpha}_{u} + ^U\overline{\omega}_u \times I_U \cdot ^U\overline{\omega}_u \\ &\quad-^U\overline{r}_{S/CU} \times ^U\overline{R}_{SU} + ^U\overline{r}_{E/CU} \times ^U\overline{R}_{EU} \\ & \quad+ ^U{\mathbf{R}}_F\left(I_F \cdot ^F\overline{\alpha}_u + ^F\overline{\omega}_u \times I_F \cdot ^F\overline{\omega}_u \right.\\ & \quad\left. - ^F\overline{r}_{E/CF} \times ^F\overline{R}_{EU}\right), \end{aligned} $$

(8)

$$ \begin{aligned} ^U\overline{T}_{Sint} & =\,^U{\mathbf{R}}_F\left(^F\overline{T}_{Enet} + I_F \cdot (^F\overline{\omega}_u \times ^F\overline{\omega}_{fu}) \right. \\ & \quad\left. + ^F\overline{\omega}_u \times I_F \cdot ^F\overline{\omega}_{fu} + ^F\overline{\omega}_{fu} \times I_F \cdot ^F\overline{\omega}_u \right. \\ &\quad \left. - ^F\overline{r}_{E/CF} \times ^F\overline{R}_{Eint}\right) \\ &\quad- ^U\overline{r}_{S/CU} \times (^U\overline{R}_{SF} + ^U\overline{R}_{Sint}) \\ &\quad+^U\overline{r}_{E/CU} \times (^U\overline{R}_{EF} + ^U\overline{R}_{Eint}), \end{aligned} $$

(9)

$$ ^U\overline{T}_{Sg} = -^U{\mathbf{R}}_F\, ^F\overline{T}_{Eg} - ^U\overline{r}_{S/CU} \times ^U\overline{R}_{Sg}. $$

(10)