An Engineering Explanation of Human Vertical Reaching: Optimal Trajectory Planning of Asymmetric Velocity Profiles in the Vertical Plane via a Softplus-Based Generalized Dynamic Jerk Cost Function.

Abstract

The results in this article demonstrate that the central nervous system CNS plans trajectories in the vertical plane by minimizing a Generalized Dynamic Jerk GJerk cost function, which we have proposed as an advancement of the traditional Minimum Jerk cost function. While the traditional Jerk cost function generates smooth trajectories by minimizing the third derivative of joint angle over time, it neglects real dynamic constraints most notably the effect of gravity. According to our engineering interpretation, the arm functions as a constrained manipulator where gravity acts as a non-linear, state-dependent bias; thus, the asymmetry in velocity profiles is not an anomaly but an optimal adaptation to the gravitational field. Consequently, it predicts mathematically symmetrical upward and downward movements, even in contexts where gravitational forces play a decisive role. The proposed GJerk cost function extends the classical framework by incorporating position-dependent dynamic coefficients into the cost function, thereby embedding gravitational effects directly within the planning process. As a result, the generated movements are not only smooth in a mathematical sense but also physically and mechanically consistent with the actual torque and energy demands imposed by the musculoskeletal system. The GJerk cost function reduces angular change in high-gravitational-torque regions because it makes the joint configuration with the highest gravitational torque more expensive within the cost function, prompting the planner to choose a slower and more conservative movement. This does not physically dissipate energy, but it acts as a behavioral damping mechanism, similar to how humans naturally slow down when approaching a high load or potentially dangerous configuration. In this way, the GJerk cost function reduces abrupt changes in joint kinetic energy and prevents sudden jumps in acceleration, ultimately producing a smoother and more stable trajectory that aligns with both biomechanical reality and observed human behavior. Within this framework, upward movements become inherently more costly than downward ones, penalizing configurations that require large torques to counteract gravity. This induces a biomechanically meaningful asymmetry, ensuring the resulting trajectories reflect the body's true mechanical constraints rather than mere geometric smoothness. Our empirical results demonstrate that the GJerk cost function provides a more realistic and physiologically grounded account of motor planning in the vertical plane, linking smoothness optimization with genuine dynamic requirements and offering a closer approximation to biological motor behavior