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Movement Curvature Planning through Force Field Internal Models Motivation Human motion studies have focused primarily on modeling straight point-to-point reaching movements. However, many goal-directed reaching movements, such as movements directed towards oneself, are not straight, but rather follow highly curved trajectories (see Fig.1). These movements are particularly interesting to study since they are essential in our everyday life, appear early in development and are routinely used to assess movement deficits following brain lesions (Petreska et al., Progress in Brain Research, 07). |
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Fig. 1. An example of the curvature of an unconstrained self-oriented movement we have recorded with motion sensors (the subject was asked to touch his nose). A, Projections of the movement in the xy-, xz-, and yz- planes. B, The velocity profile is bell-shaped, single-peaked and very similar to the velocity profile of straight point-to-point movements.C, The movement is curved in the extrinsic hand Cartesian space (left). We have performed a principal component analysis (PCA) to check whether the curvature is not an artifact of embedding the trajectory in a higher dimension. The results show that the curvature is conserved within the two principal components (right). D, The movement is curved also in the intrinsic joint angles space (left) and its two principal components (right). The joint angles represented here correspond to the three degrees of freedom of the shoulder: shoulder flexion-extension (SFE), shoulder abduction-adduction (SAA) and shoulder humeral rotation (SHR). |
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Approach We argue that curved and straight line reaching movements are generated by a unique neural controller and that the observed curvature of the movement is the result of an active control strategy that follows the geometry of one's body, for instance to avoid trajectories that would hit the body or yield postures close to the joint limits. We have developed a mathematical model that accounts for such an active control strategy and show that the model reproduces with high accuracy the kinematic features of human data during unconstrained reaching movements directed toward the head. The model consists of a nonlinear dynamical system with a single stable attractor at the target. Embodiment-related task constraints are expressed as a force field that acts on the dynamical system:
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where the first term is a damping factor proportional to the
speed of the end-effector and prevents the system from
oscillating too importantly. The second term corresponds to
an elastic force that drives the end-effector from its actual
position x(t) toward the desired target position x*(t). &alpha is a
time constant that determines the speed at which
the system slows down when approaching the target.&beta &isin
 + determines the amplitude of the speed at which the system moves globally.
g is a nonlinear function that modulates the dynamics of the
system so that it presents a typical bell-shaped velocity profile. &eta
is a multiplicative gaussian noise with zero mean and standard
deviation proportional to the distance
between the actual and desired end-effector positions,
namely ||x*(t) - x(t)||. This noise factor is the so-called signal-dependant noise (SDN) observed at the neural level of the human motor system and is necessary to initiate
the movement and to account for the intra-trials variability
in the onset of movement. Finally F(x(t)) that corresponds to the virtual force
field that encapsulates the constraints related to: a) objects in the environment that one needs to avoid (including
the subject’s body), b) dynamic properties of the human
body such as inertial properties of the limb, c) extreme joint
angles limits.
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Results We could show that not only the spatial, but also the temporal features of unconstrained and naturally curved reaching movements could be modeled through our dynamical system modulated by a virtual force-field. We found that the model was in very good agreement with kinematic data from human motions during unconstrained reaching movements directed to the head, the mean deviation observed was less than 2cm for movements of average path length superior to 1m (see Fig.2).The model is globally asymptotically stable and biologically plausible, with identified neural correlates for the model's parameters. To summarize, we suggest that embodiment should be considered as a main cause for movement trajectory curvature. |
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Fig. 2. An example of the real and modeled mean trajectories of unconstrained movements towards a target situated on the head. The shadowed trajectories represent the movement variability. |
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Applications This model was used to account for the kinematics of movements for which the subjects are specifically requested to exaggeratedly elevate the elbow. We proposed another mathematical model suggesting that movements may be planned through the combination of two concurrent controllers for the wrist and elbow in space. Coherence constraints are enforced between the two systems to simulate biomechanical constraints at the wrist, elbow and shoulder levels. External constraints, such as the presence of obstacles, are encapsulated in the virtual force which affects the planning of the movement. The predictions of the model are validated against kinematic data from human reaching motions. Four types were contrasted: intransitive versus transitive reaching motions and natural versus un-natural (elevated elbow) motions. Even though in all four movement types the movements are highly curved, the model is able to render their kinematics with high accuracy. |
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Videos
Selected publications
Journal Papers
Point-to-Point Unconstrained Gestures: Modeling Wrist and Elbow Trajectories
A. Just, B. Petreska, A. Billard, L. Craighero, A. D'Ausilio, A. Oleynik and L. Fadiga
submitted, (2008). [Detailed Record]
Movement Curvature Planning through Force Field Internal Models
B. Petreska and A. Billard
submitted, (2008). [Detailed Record]
Book Chapters
Apraxia: a review
B. Petreska, M. Adriani and O. Blanke
in Action to Cognition. Progress in Brain Research, Amsterdam : Elsevier, 2007. [Detailed Record] [PDF Format]
Posters
Modeling Unconstrained Self-Oriented Movements
B. Petreska and A.G. Billard
Presented at: Neural Control of Movement Society Meeting (NCM 2008), Naples, Florida, April 29 - May 4. [Detailed Record]
People involved in this project
