The inactivation principle: mathematical solutions minimizing the absolute work and biological implications for the planning of arm movements

Abstract

An important question in the literature focusing on motor control is to determine which laws drive biological limb movements. This question has prompted numerous investigations analyzing arm movements in both humans and monkeys. Many theories assume that among all possible movements the one actually performed satisfies an optimality criterion. In the framework of optimal control theory, a first approach is to choose a cost function and test whether the proposed model fits with experimental data. A second approach (generally considered as the more difficult) is to infer the cost function from behavioral data. The cost proposed here includes a term called the absolute work of forces, reflecting the mechanical energy expenditure. Contrary to most investigations studying optimality principles of arm movements, this model has the particularity of using a cost function that is not smooth. First, a mathematical theory related to both direct and inverse optimal control approaches is presented. The first theoretical result is the Inactivation Principle, according to which minimizing a term similar to the absolute work implies simultaneous inactivation of agonistic and antagonistic muscles acting on a single joint, near the time of peak velocity. The second theoretical result is that, conversely, the presence of non-smoothness in the cost function is a necessary condition for the existence of such inactivation. Second, during an experimental study, participants were asked to perform fast vertical arm movements with one, two, and three degrees of freedom. Observed trajectories, velocity profiles, and final postures were accurately simulated by the model. In accordance, electromyographic signals showed brief simultaneous inactivation of opposing muscles during movements. Thus, assuming that human movements are optimal with respect to a certain integral cost, the minimization of an absolute-work-like cost is supported by experimental observations. Such types of optimality criteria may be applied to a large range of biological movements.

Figure 1. Results for a simulated 1-dof upward movement, with gradient constraints on the torque.

The theoretical phase of inactivation of the muscles is shown (rectangular frame). Note that the time to peak velocity (TPV) is 0.47 in this case. It would be equal to 0.53 for the corresponding downward movement, according to experimental findings showing the same directional asymmetries. The signal u corresponds to the ratio between the net torque acting at shoulder joint and the arm's moment of inertia.

Figure 2. Results for a simulated 2-dof arm movement.

(A) Upward direction. (B) Downward direction. Torques and angular velocities, respectively noted u (N.m) and y (rad/s), are plotted with respect to time (seconds), along with the finger velocity (m/s). The successive inactivation periods at each joint and the asymmetries of the velocity profiles are clearly visible.

Figure 3. Illustration of the experimental setup.

(Left) Black trajectories show the 1-dof pointing task between targets T1 and T1′. Gray trajectories show the 3-dof experiment, starting from fully-extended arm postures (targets T1-T3′ and T1′-T3). (Right) Vertical 2-dof pointing movements, between targets T2-T2′. The position of the surface electrodes (for EMGs) and the kinematic markers is shown. Abbreviations: DA, Deltoid (Anterior); DP, Deltoid (Posterior); BI, Biceps and TR, Triceps.

Figure 4. Typical experimental data of a 2-dof arm movement performed in upward (left) and downward (right) directions.

Finger velocity profiles (upper part) and four EMGs (lower part) are amplitude normalized. The periods of muscular inactivation are emphasized by means of rectangular frames. The same abbreviations as in Figure 3 are used.

Figure 5. Typical experimental data of a 3-dof vertical arm motion performed in upward (left) and downward (right) directions.

(A) Experimental results. Finger velocity profiles (upper part) and four electromyographic signals (lower part) are amplitude normalized. (B) Simulated results. The shoulder, elbow and wrist joints were free to move. Torques and velocity are given in N.m and m/s, respectively. The solutions were computed using Pontryagin's Maximum Principle (as for the 2-dof case depicted in the Materials and Methods Section, but with more complicated formulae). Moreover, the transversality conditions of Pontryagin's Maximum Principle were necessary since the location of the target in task-space led to a set of possible terminal postures, given by a 1-dimensional manifold. The periods of muscular inactivation are emphasized by means of rectangular frames. The same abbreviations as in Figure 3 are used.

Figure 6. Typical experimental data of a 1-dof arm motion performed in upward (left) and in downward (right) directions.

Finger velocity profiles (upper part) and four electromyographic signals (lower part) are reported. Note the asymmetries of the speed profiles and the simultaneous inactivation of all muscles which occurs near the velocity peak. Data are amplitude normalized and the horizontal axis denotes time (in seconds). Same abbreviations than in Figure 3. The same abbreviations as in Figure 3 are used.

Figure 7. Simulated fingertip paths in the 2-dof case.

(A) Finger trajectories for different movements toward targets located on a circle. Initially the finger position is at the center of the circle. For more details about the task and to compare the results, see ,. (B) Finger trajectories for four different movements performed in the sagittal plane (T1 to T5, T2 to T6, T3 to T7, and T4 to T8). For more details about the task and to compare the results, see .

Figure 8. Mechanical model of the 2-dof human arm.

The subscripts 1 and 2 denote the shoulder and elbow joints respectively. Generalized coordinates θ, joint torque τ, moment of inertia I, segment mass m, segment length to the center of mass lc, and gravity acceleration g are denoted.

Figure 9. Intuitive illustration of the Inactivation Principle proof.

Figure 10. Different optimal strategies in the 1-dof case, depending on the movement duration T.

The strategy S1 depicts the fastest movement w.r.t. the bounds imposed on the control. Strategy S2 was depicted in Figure 1 with gradient constraints on the control u. Strategies S3, S4, and S5 show inactivation phases (as well as S2). An inactivation phase corresponds to the period where the control signal u is zero. When T becomes large (T≥0.6 s in this case), the inactivation disappears (S6 and S7 strategies) according to experimental findings. The angular position and velocity and the control signal are given in radians, rad/s, and rad/s², respectively. Note that the control signal u corresponds to the ratio between the net torque acting at shoulder joint and the arm's moment of inertia.

Figure 11. Phase portrait for p ₀≥k in the plane (z,w).

The bisector (z = w) corresponds to the set of velocities equal to zero. The upper and lower semi-plane corresponds to positive and negative angular velocities, respectively. An optimal path starts and ends on this line. This figure illustrates the optimal phase portrait corresponding to the S2 strategy (for an upward motion). Regions are denoted by boxed numbers and the commutation times correspond to switches between regions. For instance Region 5 corresponds to the inactivation region (i.e., the control signal is zero here). Note that the different strategies illustrated in Figure 10 are easily understood with this phase portrait, since optimal paths may start and end in different regions. The constants k, α_U, α_D, and u ⁺ and u ⁻ are parameters depending respectively on the mechanical model of the arm, the coefficients involved in our cost function, and the boundary values imposed on the control u.

Figure 12. Optimal Triphasic Pattern.

Illustration of the optimal behavior of a 1-dof arm, under the small angles assumption and with a pair of agonistic and antagonistic muscles, modeled by first-order dynamics. The subscripts 1 and 2 denote the flexor and extensor muscles, respectively. The triphasic pattern is an agonistic burst, followed by an antagonistic burst, and again an agonistic burst. The inactivation occurs between the first agonistic and antagonistic bursts. The times t_i denote the commutation times. The left graphs illustrates the behavior of the angular torques (u). The right graphs illustrate the behavior of the control signals (ν), that are the input signals for muscles contractions (i.e., the signals driven by motoneurons). All signals are plotted with respect to time t varying between 0 and T.