Movement is essential to human wellbeing. However, the control of movement is a very difficult problem. To produce deft and robust movements, the human nervous system must continuously control over 600 muscles while handling nonlinearities, nonstationarities, delays, noise, and uncertainties (Franklin and Wolpert, 2011). Despite these difficulties, humans move with apparent ease. However, human motor capability may become impaired due to age, illness, or injury. Robotic systems are also faced with many of the same challenges (Egeland et al., 1991; Park, 2002; Guigon et al., 2007; Peters et al., 2009), but meet with much less success than their healthy human counterparts (Yang et al., 2011; Vanderborght et al., 2013). A better understanding of the roles that the nervous and musculoskeletal systems play in producing movement will likely lead to advances in rehabilitation and robotic control.

Many neuromechanists employ simple tasks to study the nervous system in action under controlled conditions. Isometric tasks in which subjects interact with an immoble force sensor and reaching tasks in which the hand is moved from one point to another are commonly used to study sensorimotor learning (Rotella et al., 2015), movement control (Fitts, 1954), and neurophysiology (Shadmehr and Krakauer, 2008) in the upper extremity. Subjects may also be asked to interact with robotic co-workers that can record reaching dynamics (Burdet et al., 2001), generate disturbances, or create force fields (Shadmehr and Mussa-Ivaldi, 1994) during these tasks. When carefully considered, these experiments can provide a wealth of information on how the nervous system controls movement. However, these tasks are greatly simplified from real-world tasks. To study more complex tasks, some researchers have developed simple games, such as conkers, to study sensoriomotor learning (Sternad et al., 2011). However, even these studies simplify real-world tasks and rarely consider certain features of real-world tasks such as impacts.

Despite many possible ways to perform most tasks (Bernstein, 1967), upper extremity movements are highly stereotyped. Researchers note consistent characteristics such as bell-shaped velocity curves (Hollerbach and Atkeson, 1987; Berardelli et al., 1996) and speed-accuracy tradeoffs characterized by Fitts' Law (Fitts, 1954; Bootsma et al., 2004; Zhai et al., 2004). Fitts' Law expresses the time to complete a reach as a logarithmic function of the size of the target and the distance to the target (see Equation 1). In experiments relating to Fitts' Law, the kinematics (the beginning and final position of the arm or cursor) are prescribed and the subject is left to determine the time to reach. In certain periodic movements however, the time to complete an upper extremity movement can be specified and the subject left to determine the kinematics.

Movement is constantly refined by biological processes such as learning and evolution (Todorov, 2004). Because of this constant refinement, many researchers note that optimal control models utilizing cost functions such as minimum variance (Harris and Wolpert, 1998), minimum effort (Crowninshield and Brand, 1981), minimum jerk (Flash and Hogan, 1985), and minimum torque change (Uno et al., 1989) can be excellent models for the nervous system. In fact, many of the observed stereotypical behaviors discussed in the previous paragraph can be explained by optimality principles. Optimal control models have been used to reproduce human-like behaviors such as reaches (Todorov and Li, 2005), walking (Anderson and Pandy, 2001), and jumps (Anderson and Pandy, 1999; Ong et al., 2016). Though occasionally studied (Côté et al., 2008; Müller and Sternad, 2009), one activity that remains conspicuously unmodeled is Bernstein's hammering task (Bernstein, 1967; Müller and Sternad, 2009) that inspired much research into motor control and learning.

Here we extend Bernstein's hammering task into a targeted periodic impact task. We recorded impact forces and upper extremity kinematics in hammering. In order to examine how hammering strategies might change with different conditions, we used a set of hammers with different physical properties (hammer face area and mass) and prescribe different hammering frequencies. We hypothesized that hammering impact velocity and maximal height attained are the result of a tradeoff between maximizing task performance (quantified here as a maximal impact velocity) and minimizing effort (Crowninshield and Brand, 1981; Nelson, 1983). In order to test whether the mechanics of this task adhere to current theories in optimal human motor control, we implemented a feedforward optimal controller (Todorov, 2004) on a planar torque-driven 3-segment dynamical model of the upper extremity holding a hammer (Figure 8) using model parameters from Winter (2009). Our results show that humans appear to select optimal impact velocities that reflect a tradeoff between accomplishing the task and minimizing effort that do not adhere to Fitts' Law.

The goal of this study was to examine the mechanics of a human upper extremity impact task and determine whether existing models of upper limb movement can explain the data. We found that subjects plan optimal trajectories that are a tradeoff between maximum impact velocity and minimal effort reminiscent of Fitts' Law and that are robust to different hammer conditions. However, we found that an altered version of Fitts' Law was able to better match the data than the typical formulation. We also found that end-effector speeds follow a “bell curve and a half” trajectory in hammering in which the hammer head moves upwards with a bell-shaped speed profile and then downwards with a bell-shaped profile before being truncated before the zenith of the curve (Figure 3).

Our analyses showed that Fitts' Law can be applied to human hammering (R² ≥ 0.88). However, the large R² values belie an apparent discrepancy between the curves generated using Fitts' Law and the experimental results (Figure 7, gray lines). Therefore, we identified a relationship between movement time and target distance that better reproduces the experimental data (Equation 4, R² ≥ 0.99). In most Fitts' Law experiments, subjects are prescribed a reaching distance and are asked to move as fast as possible (Fitts, 1954). However, in our experiments, we constrain permitted movement time using the metronome and subjects were allowed to select the distance to reach. This difference may account for the relative effectiveness of our inverted formulation of Fitts' Law. However, other previous studies have reported violations of Fitts' Law (Adam et al., 2006; Glazebrook et al., 2015). Glazebrook et al. (2015) determined that these Fitts' Law violations are the result of pre-planning of movements. This explanation is also certainly plausible in the context of a cyclical task such as hammering. Finally, several studies have noted that Fitts' Law does not hold for movements in which subjects were not asked to move as quickly and as accurately as possible (Young et al., 2009). We do not explicitly instruct our subjects to move as quickly and accurately as possible. Instead, we instructed them to accomplish a task that is directly dependent on the speed of the movement and allow them to balance that movement speed with their motor capability, which we believe to be an approximation of the instructions to move as quickly and as accurately as possible. In hammering frequencies above 1 Hz, the computed values of α indicate that subjects weight movement speed very highly (Figures 2, 3), and thus likely approach a fast-as-possible movement for which Fitts' Law is presumed to be valid.

Our feedforward optimal hammering simulation was able to reproduce many of the features of human hammering (Figures 2, 3). Our simulations also allow us to show that humans prefer to emphasize energy transfer to the nail (larger values of α) when task constraints are high (high hammering frequencies) and minimal efforts (smaller values of α) when task constraints are low (low hammering frequencies; Figure 8). Our cost function was formulated to minimize the sum squared actuator effort, which serves to keep commanded joint torques small. These small actuation signals prevent excessive energy expenditure during the task (Crowninshield and Brand, 1981; Missenard and Fernandez, 2011), but this quadratic formulation might also serve to keep disturbances from motor noise whose effects are multiplicative with actuator effort small (Harris and Wolpert, 1998; Todorov and Li, 2005; Franklin and Wolpert, 2011). In this context, the adaptive prioritization that we observed (changing values of α) might be due to fewer task constraints permitting higher peak heights to be attained at slow hammering frequencies, thus increasing the potential for errors to accrue and increasing the relative importance of accuracy. While the exact cost function used by the nervous system cannot be known exactly, the current formulation reproduces many of the features observed in the experimental results including maximum heights attained, the general trajectories followed, and the robustness to different hammers (similar values of α for different hammers at the same hammering frequencies). However, this model failed to capture the latency after impact before initiating the upward movement of the hammer. This discrepancy may be due to compliance in the musculoskeletal system (e.g., series-elastic muscle-tendon units, Hill, 1938; Fung, 2013) that was not captured by our model.

Despite different hammer dynamics (Table 1), hammering kinematics were fairly uniform across many different cases with few statistical differences found between the different hammers in the time from maximal height to impact, maximal hammer height, and impact velocity. Previous studies have suggested that the redundancy of the human musculoskeletal system (Bernstein, 1967) may contribute to considerable robustness to slight changes in dynamics (Martelli et al., 2015; Simpson et al., 2015) or to dysfunction (Arnold et al., 2005; Hicks et al., 2008; Correa et al., 2012; Steele et al., 2012). While these studies rely on highly redundant lower body musculoskeletal models, other studies examining less redundant body parts have shown limited ability to compensate for dysfunction (Valero-Cuevas and Hentz, 2002; Kutch and Valero-Cuevas, 2011). However, detailed models of the upper extremity indicate muscular redundancy on the same level as detailed models of the lower body (Table 4) suggesting that similar robustness to perturbations might be expected. The human nervous system may also select control strategies that are purposefully robust (Mitrovic et al., 2010; Franklin and Wolpert, 2011), but our formulation does not include any such criteria, suggesting that consistent movement patterns across conditions might be due to embodied intelligence (e.g., redundant actuators and compliance).

Stiffness, or impedance, is a crucial parameter modulated by humans to stably interact with their environment (Burdet et al., 2001; Franklin and Wolpert, 2011). Impedance is difficult to record experimentally, but previous studies have attempted to estimate joint stiffnesses based on muscle properties (Hu et al., 2011), through simulation (Thelen et al., 2003), or by experimentally recording endpoint stiffnesses (Burdet et al., 2000, 2001). Because of practical limitations, measurements of muscle activity or impedance were not included in this study, but likely play an important role in impact tasks and should be considered in future works.

Despite the difficulty of controlling a highly nonlinear plant using noisy control signals and noisy sensors with variable delays in an uncertain environment, biological movement appears to be highly robust. However, robustness has not been well addressed in robot learning (Schaal and Atkeson, 2010; Nguyen-Tuong and Peters, 2011) primarily because it is difficult to design controllers that are robust to the model structure or parameter errors. One possible solution is to use control policies with optimization criteria based on biological models. For example, the tradeoff between maximizing task performance and accuracy could potentially serve as an optimization criteria for robot hammering.

In this paper, we have extracted the mechanics involved in a targeted upper extremity impact task and demonstrated that the human motor control strategies involved are robust to many different conditions including hammer mass, hammer face area, and timing constraints. We have shown that while many traditional models of human reaching hold for this novel task (bell-shaped speed profiles and Fitts' Law), an altered version of Fitts' Law can better match experimental results. We have also demonstrated that optimality principles previously demonstrated for reaching movements can be generalized to targeted impact tasks and thus lay a framework that can be used for the planning of targeted impact tasks in robots.

TP, CS, AU, and AI contributed to the design, execution and drafting of this work, and approved the final manuscript. Experimental data was collected and analyzed by TP and CS.