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Kinetic Rules Underlying Multi-Joint Reaching Movements. Daniel M Corcos†, James S. Thomas*, and Ziaul Hasan†. School of Physical Therapy*, Ohio University, Athens OH. School of Kinesiology†, University of Illinois at Chicago, Chicago IL 642.4

IntroductionAccording to Bernstein (1967) the CNS may resolve the problem of

kinematic redundancy by reducing the independent degrees of freedom in a coordinated task. Evidence of coupling of the limb segments have been reported for gait activities (Bianchi et al., 1998; Borghese et al., 1996), for trunk flexion tasks (Alexandrov et al., 1998), and recently for whole-body reaching tasks (Hasan & Thomas, 1999). It has also been reported that the CNS may reduce the complexity of multi-joint tasks by using similarly shaped kinematic waveforms amongst the limb segments (Alexandrov et al., 1998). It is possible, however, that the problem of kinematic redundancy may actually be resolved at the kinetic level. The “linear synergy” hypothesis states that the observed kinematic patterns are simply emergent properties of the mechanical system (Gottlieb et al., 1997; Gottlieb et al., 1996a; Gottlieb et al., 1996b). Others have also shown that kinetic parameters are optimized in various reaching tasks (Kawato, 1996; Soechting et al., 1995; Nishikawa et al., 1999). From this perspective inter-joint coordination rules are kinetically based.

While it has been proposed that the CNS uses a torque based approach to plan movement patterns,and clear evidence provided for 2 DOF reaching tasks, it is not clear the concept of a “linear synergy” would hold for a 6 DOF reaching task, and if a set of torque based rules governing these tasks could be established.

MethodsThe time-series changes in orientation of the forearm, humerus, trunk, pelvis,

thigh, and shank were measured in 10 male and 10 female subjects performing whole-body reaching tasks to two target locations determined by each subject’s limb segment lengths (Figure 1). Subjects reached for the targets at two speeds (self –selected and fast paced), and from three initial positions (i.e. initial center of pressure at a self-selected position, directly over the malleolus, and 6 cm anterior to the lateral malleolus). The subjects were given no instructions on the limb segment geometry to use while performing these reaching tasks.

The dynamic components of the joint torques were derived from a sagittal plane inverse dynamics analysis in which the gravitational component was removed. A linked-segment model consisting of seven segments was used for the calculations.

The time-series dynamic joint torques were analyzed by principal component (PC) analyses to determine if there was a commonality amongst the shapes of the waveforms across segments, across trials, and across subjects. Principal component analyses were also performed on the scaling of the kinematic waveforms across the experimental manipulations, to probe for any lawful relationships amongst their relative magnitudes.

ResultsThe PC analyses revealed that the time courses of the dynamic joint torques

were nearly identical in shape within a single trial (Figure 4A-D), across trials (Figure 5A), and across individuals Figure 5B). The PC analyses of each of 480 movement trials (24 movement trials X 20 subjects) revealed that on average, 98.3% of the total variance of the dynamic joint torques was accounted for by the 1st principal EigenCurve. A PC analysis of the 6 scaling coefficients from the 24 movement trials revealed that for 19 out of 20 subjects a linear relationship amongst the excursions of each segment was observed across task manipulations (i.e. greater than 95% of the total variance of the scaling coefficients were accounted for by single EigenVector. See Figure 6). Additionally, the ratios of the 6 dynamic joint torques were invariant to manipulations of movement speed, and initial posture.

Trunk Length

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30° Hip Flexion

Target 1: Low Target Target 2:High Target

ConclusionsKinetic Rules: Based on the PC analyses of the dynamic joint torques of individuals performing reaching tasks that necessitate some forward bending of the trunk we have developed the following kinetic rules for these multi-joint reaching tasks.

Kinetic Rule 1a. Within any given movement trial of a multi-degree of freedom reaching task, the shapes of the sagittal plane dynamic joint torque waveforms are nearly identical.

Kinetic Rule 1b. There is one common waveform that describes the dynamic joint torques in the sagittal plane amongst all the segments used in a multi-degree of freedom reaching task that is valid for all subjects across target locations, movement speeds, initial COP conditions, and gender.

Kinetic Rule 2. There is a linear relationship amongst the scaling coefficients of the dynamic joint torques waveforms for multi-degree of freedom reaching tasks.

Kinetic Rule 3. The ratios of the scaling coefficients amongst the segments of a multi-degree of freedom task are invariant to task manipulations of movement speed and initial posture.

Figure 1 Target locations were determined from the subject's hip height, trunk length and arm length . Subjects could reach the low target, in theory , by flexing their hips 60° (with the elbow extended and the shoulder flexed 90°) without any motion of the ankle, knee, or spine.

Figure 3 The 6 time series dynamic joint torques from a single movement trial ( high target, fast movement speed) have been amplitude normalized to unity and separated on the plot for visual clarity.

Figure 4 A) The time series dynamic joint torques of the ankle, knee, hip, abdomen, shoulder, and elbow joints of an individual subject reaching for the high target at a comfortable speed. B) The 1 st principal EigenCurve and the 6 scaling coefficients derived by PC analysis. The 1 st principal EigenCurve accounted for 97.1% of the total angular variance in this trial. C) Time series dynamic joint torques of the same subject reaching for the high target at a fast paced speed. D) The 1 st principal EigenCurve for the fast trial accounted for 97.2% of the total angular variance

Figure 5 A) The time normalized 1st Common EigenCurves from the 20 subjects. On average the Common EigenCurve accounted for 95.6% of the total variance of the dynamic joint torques from the 24 movement trials of each subject. B) The 1st Global EigenCurve which was derived from the time normailzed 1st principal EigenCurves from each trial (24) of every subject (20). The 1 st Global EigenCurve accounted for 88.4% of the total angular variance across the 480 movement trials.

Figure 6 The scaling coefficients for the ankle, knee, and hip dynamic joint torques for the 24 movement trials of an individual subject are plotted in 3 dimensional space. Each circle represents one movement trial. The scaling coefficients for the 24 movement trials are plotted as circles. The green vector represents the common EigenVector (i.e. the best fit line through the scaling coefficients of the 24 movement trials). The black vector runs from the origin of the three-dimension space to the mean of the scaling coefficients from the 24 movement trials (hollow blue circle). To determine if the common EigenVector ran through the origin, we calculated the angle between the common EigenVector (green vector) and the black vector. For this subject the angle between these vectors was 9.4 in 6-D space. The open circles are from trials where subjects reached for the low target, and the filled circles represent reaches to the high target. The red triangles represent the scaling coefficients for the 8 movement trials with the instructed limb segment geometry.

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Figure 2 A subject reaching for the low target at a self-selected pace.