Different Predictions by the Minimum Variance and Minimum Torque-Change Models on the Skewness of Movement Velocity Profiles

Abstract

We investigated the differences between two well-known optimization principles for understanding movement planning: the minimum variance (MV) model of Harris and Wolpert (1998) and the minimum torque change (MTC) model of Uno, Kawato, and Suzuki (1989). Both models accurately describe the properties of human reaching movements in ordinary situations (e.g., nearly straight paths and bell-shaped velocity profiles). However, we found that the two models can make very different predictions when external forces are applied or when the movement duration is increased. We considered a second-order linear system for the motor plant that has been used previously to simulate eye movements and single-joint arm movements and were able to derive analytical solutions based on the MV and MTC assumptions. With the linear plant, the MTC model predicts that the movement velocity profile should always be symmetrical, independent of the external forces and movement duration. In contrast, the MV model strongly depends on the movement duration and the system's degree of stability; the latter in turn depends on the total forces. The MV model thus predicts a skewed velocity profile under many circumstances. For example, it predicts that the peak location should be skewed toward the end of the movement when the movement duration is increased in the absence of any elastic force. It also predicts that with appropriate viscous and elastic forces applied to increase system stability, the velocity profile should be skewed toward the beginning of the movement. The velocity profiles predicted by the MV model can even show oscillations when the plant becomes highly oscillatory. Our analytical and simulation results suggest specific experiments for testing the validity of the two models.

Optimal Control of Natural Eye-Head Movements Minimizes the Impact of Noise

“…In contrast to the minimum-effort approach (Kardamakis and Moschovakis, 2009), which explicitly penalizes eye eccentricity with an additional penalty function, in our model the initial position dependence is a consequence of the plant dynamics. In accordance with Harris and Wolpert (1998) and Tanaka et al (2004), our model also accounts for the skewed eye and bell-shaped head velocity profiles (Fig. 5).…”

The Sources of Variability in Saccadic Eye Movements

“…Importantly, SDN in motor commands is a relatively small source of this variance. This is a serious departure from the commonly made assumption (Harris and Wolpert, 1998;Tanaka et al, 2004Tanaka et al, , 2006) that endpoint variance results from signal-dependent noise only. It remains to be determined which trajectories minimize the consequences of the actual noise.…”

“…The properties of the motor unit pool of a muscle, such as recruitment, are such that the force produced by a muscle also has SDN (Jones et al, 2002;Hamilton et al, 2004). SDN is zero-mean, white Gaussian noise in the magnitude of the signal with an SD proportional to the magnitude of the signal, and it has been used extensively in modeling studies (Harris and Wolpert, 1998;Tanaka et al, 2004Tanaka et al, , 2006. We added SDN to the motor commands at each time-step of the mean trajectories.…”

A mathematical analysis of the minimum variance model of human-like reaching movements

“…, α 3 = b0+b I0 + 1 ta + 1 te , The system parameters are set as follows [13], [22]: the arm's moment of inertia I 0 = 0.25kg · m 2 , the intrinsic viscosity coefficient b 0 = 0.2kg · m 2 /s, the intrinsic stiffness coefficient k 0 = 0N/m, L 0 = 0.35m, the times constants are t a = 0.04s, t e = 0.03s. The externally applied viscosity b and stiffness k are zero.…”