An Experimentally Confirmed Mathematical Model for Human Control of a Non-rigid Object

Determining the principles used to plan and execute movements is a fundamental question in neuroscience research. When humans reach to a target with their hand, they exhibit stereotypical movements that closely follow an optimally smooth trajectory. Even when faced with various perceptual or mechanical perturbations, subjects readily adapt their motor output to preserve this stereotypical trajectory. When humans manipulate non-rigid objects, however, they must control the movements of the object as well as the hand. Such tasks impose a fundamentally different control problem than that of moving one’s arm alone. Here, we developed a mathematical model for transporting a mass-on-a-spring to a target in an optimally smooth way. We demonstrate that the well-known “minimum jerk” model for smooth reaching movements cannot accomplish this task. Our model extends the concept of smoothness to allow for the control of non-rigid objects. While our model makes some predictions that are similar to minimum jerk, it predicts distinctly different optimal trajectories in several specific cases. In particular, when the relative speed of the movement becomes fast enough, or when the object stiffness becomes small enough, the model predicts that subjects will transition from a uni-phasic hand motion to a bi-phasic hand motion. We directly tested these predictions in human subjects. Our subjects adopted trajectories that were well-predicted by our model, including all of the predicted transitions between uni-phasic and bi-phasic hand motions. These findings suggest that smoothness of motion is a general principle of movement planning that extends beyond the control of hand trajectories. INTRODUCTION Determining the underlying principles humans use to plan and execute movements is a vital question in motor neuroscience research (Engelbrecht 2001; Gomi and Kawato 1996; Harris and Wolpert 1998; Wolpert and Ghahramani 2000). Much attention has focused on how humans make point-to-point reaching movements with their hands or manipulate simple rigid objects. When adapting to various perceptual and mechanical perturbations, humans eventually return to making stereotypical smooth movements (Bock 1993; Dizio and Lackner 1995; Flanagan and Lolley 2001; Flanagan et al. 1999; Flanagan et al. 2003; Shadmehr and Mussa-Ivaldi 1994; Wolpert et al. 1995a). These previous experiments artificially manipulated different parameters (e.g. inertia, stiffness, damping, etc.) of the arm. Such manipulations constitute parametric perturbations to the control system because they do not alter the fundamental structure of the arm (e.g. the number of joints, muscles, etc.). Therefore, they do not alter the number or type of state variables used to characterize the arm’s equations of motion. These equations of motion remain effectively unchanged, except for the values of one or more of their parameters (e.g. mass, stiffness, damping, etc.). Accordingly, such parametric perturbations do not require any changes in the movement kinematics required to accomplish a given task. The exact same kinematics can be achieved by adjusting the profile of forces and torques being input into the system. This helps explain why humans learn to regain their original movement patterns despite these perturbations. However, humans often perform tasks involving non-rigid objects, such as a cup of coffee or a briefcase with a hinged handle. In these tasks, the goal is to impose some particular motion on the object (e.g. the coffee in the cup), rather than the hand. Manipulating non-rigid objects poses a fundamentally different control problem than that of moving one’s arm (Huang et al. 2002; Lynch and Mason 1999; Schaal and Atkeson 1993). Copyright (c) 2003 by the American Physiological Society. Articles in PresS. J Neurophysiol (November 5, 2003). 10.1152/jn.00704.2003 Optimally Smooth Transport of Non-Rigid Objects pg. 2 of 14 New state variables (i.e. the position and velocity of the object) are added to the “arm-plus-object” system being controlled. This then fundamentally alters the structure of the governing equations of motion: entirely new equations must be defined for each new state variable. Such tasks therefore constitute structural perturbations to the system being controlled. Consequently, limb motions that were optimal for the unperturbed system may no longer work for the perturbed system (Lynch and Mason 1999). Although one could still adjust their input forces and torques accordingly to achieve the previously optimal motion of the hand, this may result in a completely irrelevant or even unwanted motion of the object. Achieving the desired motion of the object requires learning to apply the appropriate sequence of input forces to the object (Dingwell et al. 2002). Furthermore, humans cannot control the motions of non-rigid objects directly because those objects are not directly acted on by our muscles. The motion of the object can only be controlled indirectly through our interaction with the dynamics of the object itself. Given these fundamental differences between parametric and structural perturbations and their implications, it is important to determine whether or not the underlying principles believed to be involved in the planning and execution of reaching movements can be extended to tasks involving the purposeful manipulation of complex dynamic objects. It has been argued that to fully validate any theory intended to describe how movements are planned and executed requires that we apply those theories to so-called “critical tests” involving tasks that are substantially different than the tasks for which those theories were originally formulated (Engelbrecht 2001). We believe tasks involving structural perturbations to the control system, like the one described in the present study, may provide just such a critical test. In particular, the present study was conducted to determine if the principle of achieving optimum smoothness remains a viable goal for motion planning when manipulating complex dynamic objects and to determine what kind of hand motions are required to ensure smooth movements of these objects. Here we formulate a testable hypothesis, optimally smooth transport (OST), whereby the controller derives the motions of the hand that achieve the maximally smooth motion of the transported object, compatible with the constraints of the hand/object interaction. Our OST model is a generalization of the minimum-jerk principle for reaching (Flash and Hogan 1985) where now the optimization criterion is applied to the object rather than the hand. We applied this principle to the task of transporting a mass on a spring to a target. We demonstrate that this task violates the originally proposed minimum jerk principle in the sense that the task can not be achieved by producing either a minimum jerk movement of the hand or a minimum jerk movement of the mass. Our extended OST model of smoothness overcomes these limitations in the original theory. The OST model predicts some movement contexts in which the optimal solutions are similar to those of minimum-jerk and other contexts in which the solutions are very different. Subjects adopted trajectories that were well-predicted by our model under all of these different contexts. Our results demonstrate that subjects learned to alter their hand trajectories accordingly under each test condition so as to achieve an optimally smooth motion of the object. These findings suggest that the central nervous system achieves near optimal smoothness when planning the movements of the end point (in this context, the mass) in a general sense. METHODS The Task The task we studied was that of transporting a mass-on-a-spring attached to the hand from one point to another in the horizontal plane. The equations of motion for this object were: