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Table of contents

In this research, we are not focused on the source of the sensory information; we assume that the current biomechanical states posture and velocities are available to the CNS when necessary. This assumption seems to be valid for healthy individuals, as a wide range of sensory organs is available to sense and transmit information to the CNS.

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However, a pathological condition might limit the CNS access to this available sensory information. For instance, in a deafferented patient, the sense of position and therefore the motor skills is largely lost due to the loss of somatosensory inputs Bringoux et al. This paper is organized as follows. In the Method section, the experimental procedure, the planar arm model, the nonlinear model predictive controller, and forward dynamic simulation framework are provided.

Next, in the Results and Discussion section, the use of NMPC as the motor control unit in human reaching tasks is presented and discussed. This study investigates the use of anticipatory planning with continuous error correction by the CNS during reaching tasks. The first goal of this study is to study the effects of varying the prediction horizon on the hand trajectory and muscle activities in a reaching task.

Secondly, the capability of the NMPC as an optimal feedback controller for tracking predefined trajectories has been investigated. Lastly, the effectiveness of the proposed NMPC for the simulation of reaching to moving targets is studied. Finally, Conclusions and Future Work are presented. To examine the accuracy of the NMPC predictions, a 27 year old male subject was selected to perform reaching tasks.

In these experiments, the subject was seated with the arm elevated at the shoulder level. The subject was asked to move his hand from an initial central position to one of eight final targets spread evenly on a circle of 20 cm radius at a self-selected convenient speed. The experiment was repeated 10 times for each target with 2 min rest intervals between each set. The subject also performed reaching to moving targets. He was instructed to reach to a target, which was relocated to another position midway through the movement. The subject was instructed to adjust his motion to reach to the moving target.

The subject was also given 5 min to rest before performing the reaching moving targets experiment. The experiments have been approved by the Office of Research Ethics at the University of Waterloo and carried out with written informed consent from the subject. The subject gave written informed consent in accordance with the Declaration of Helsinki. In this research, a planar arm model similar to the one developed by Ghannadi et al. The model consisted of torso, upper arm, and forearm to simulate the hand motion.

The torso was fixed and the shoulder and elbow were modeled using revolute joints. A modified Hill-type muscle model with muscle excitation-to-activation dynamics was used to simulate the skeletal muscle contraction dynamics see Appendix A for details. The muscle parameters of the planar arm model i.

These parameters were tuned through a series of optimizations so that the planar model provides the same joint torques as a high-fidelity three-dimensional upper extremity model Ghannadi et al. The details of how the upper-extremity muscle groups are lumped into these representative muscles can be found in Ghannadi et al.

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The arm motion is controlled by complex commands descending from the CNS, which are the combination of the motion prediction feedforward control from an internal representation of the body and environment or so-called internal model; Desmurget and Grafton, , and the corrective command from the sensory organs to correct any errors due to uncertainty or unknown environment feedback control. This complexity is captured here by a model-based NMPC with a receding horizon. The NMPC uses a control-oriented model COM representing the human's internal model to predict the optimal trajectory, and feedback information to correct the prediction errors.

The state variables at the current time t o are obtained from the current sensory information. At the new time step, the optimal solution is re-evaluated with the new initial conditions for the receding horizon and iteratively applied to the system. By incorporating the feedback information, the NMPC is converted from a completely open-loop controller to an optimal closed-loop controller. The NMPC can handle constraints on both the states and the inputs. In musculoskeletal models, the muscle activation command must be non-negative and less than one, and constraints on states can be added to avoid unphysiological movements.

The solid lines shows the optimal muscle activation and state trajectories in the given prediction horizon. B Hand position trajectory. The hand moves from its original position to the marked position on its left. In this article, the optimal dynamics over the prediction horizon were calculated using the GPOPS-II optimal control package that utilizes an orthogonal collocation method Patterson and Rao, This method is a direct simultaneous optimization method in which both states x and inputs u are parameterized using a series of connected Legendre polynomials and become part of a Nonlinear Programing NLP problem.

Here, the arm model described in the previous section was used as the COM of the NMPC and the simulation model for planar reaching and pointing tasks. The dynamic equation of the arm model can be described by:. Therefore, the NMPC cost function shown in 1 is converted to:. The hand position is calculated from. The physiological cost G M u t is defined as:.

The term G M in the cost function represents the neural excitation effort to perform the reaching tasks. An hp-adaptive mesh refinement method Liu et al. Then, the first five-percent e.

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The new position and orientation of the arm are measured and sent back to the NMPC as initial conditions of the next iteration. In this research, we have assumed that the sensory organs can measure the exact joint angles and angular velocities. Uncertainty can be added to the measurements to account for the noise within the sensory organs, and to simulate the variability in the movement repetition. However, this has not been included in the scope of this work.

The optimal muscle activations are shifted and considered as the initial guess of the next iteration. Unlike the NMPC simulations, which continue until the position tracking error passes a certain threshold, the final simulation time and the final position of the hand are explicitly specified in the DO simulations. The DO cost function is:. The same physiological cost function as in the NMPC simulations Equation 6 was used in DO to compute the optimal hand position trajectory and muscle activations. In this section, a goal-directed reaching task is simulated and the effect of prediction horizon length variation on the hand trajectory and muscle activation is studied.

This task was simulated using NMPC with 0. The DO final simulation time was chosen to be 1. Despite the slight differences between the trajectories, the solutions with 0. The Pearson factor between the DO and experiment is 0. This signifies the importance of prediction horizon; with short prediction horizons, the controller is more cautious and takes longer to reach a desired position.

The reaching error at 1. As expected for this motion, the shoulder mono-articular flexor muscle 1 , elbow mono-articular flexor muscle 3 and bi-articular flexor muscle 5 are activated at the beginning to accelerate the body; then, the antagonistic muscles are activated to reach a full stop at the desired position. A Shoulder angle variation during the reaching task, B Elbow angle variation during the reaching task. Optimal activations of flexor and extensor muscles: A Optimal activation of shoulder mono-articular flexor muscle 1 , B Optimal activation of elbow mono-articular flexor muscle 3 , C Optimal activation of bi-articular flexor muscle 5 , D Optimal activation of shoulder mono-articular extensor muscle 2 , E Optimal activation of elbow mono-articular extensor muscle 4 , F Optimal activation of bi-articular extensor muscle 6.

In the DO, similar to NMPC, flexor muscles are active at the beginning of the motion to accelerate the hand toward the target, then the extensor muscles are activated to stop the hand movement a bang-bang control strategy. Since DO has to stop at the specified final time 1. On the other hand, the flexor muscle activities at the accelerating phase of the simulation are larger for NMPC than DO because the trajectory error at the beginning of the motion is large and exponentially reducing when it gets closer to target.

It can be observed that as the prediction horizon increases, the NMPC predicts larger muscle activities at the beginning of the motion. The correlation coefficient for extensor muscles are not reported since the EMG activity of these muscles were minimal in the experiments.

Human Motion Simulation 1e

As suggested by Morasso , subjects tend to move in straight lines with bell-shaped tangential-velocity profiles when reaching for a target. In these simulations, the NMPC tends to accelerate the hand more quickly than the DO; in DO, the optimization knows the final time and can distribute the acceleration over a longer time. On the contrary, in the receding horizon NMPC, the controller can only predict the motion as far as the prediction horizon.

This results in the fast acceleration at the beginning the motion due to large tracking errors and slow deceleration at the end of motion due to small tracking errors. A Hand speed vs. In DO simulation, the final time is specified as 1. B Hand speed vs. The hand moves to a target 20 cm to the left of its initial position. Reaching to eight different directions was also simulated using NMPC with 0. In the NMPC simulations, a smooth 5th-order polynomial with zero initial and final velocities and accelerations is used as the desired straight-line hand trajectory.

These trajectories begin at the initial position of the hand and end at the target positions.


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We have assumed that the CNS plans a trajectory to reach a target and constantly monitors the deviations from this trajectory and the target position. In this section, we study the case where the target position is suddenly relocated. Then 1 s later, the target position suddenly moves to the point B. This protocol was achieved in the lab by manually moving the target from its initial point A to B when the subject reached half the way to A. In this simulation, the time delay related to the visual cognition of this change [about ms, Jeannerod, ] has not been considered.

It is not possible to simulate this scenario with DO; it is one of its disadvantages compared to NMPC, which is able to make online adjustments to the hand trajectory. In this research, we presented a NMPC to mimic the human motor control system. The results showed that it can successfully replicate certain features of human motor control such as path planning and target tracking.


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This controller is a fully predictive optimal controller that simultaneously solves the kinematic redundancy and muscle-sharing problem. In hierarchical models Menegaldo et al. In the first step, an optimal trajectory is generated based on a kinematic criterion; then, in the second step, the muscle sharing problem is solved based on another criterion kinetic criterion.

The main drawback of hierarchical models is that they follow a preplanned trajectory output of the first step ; therefore the online movement corrections are in favor of trajectory tracking. However, the NMPC controller simultaneously takes into account both kinetic and kinematic exertions to determine an optimal path along with the optimal muscle activations to achieve it.