[Paper Review] Minimum Jerk Trajectory Generation for Straight and Curved Movements: Mathematical Analysis

Abdel‐Nasser Sharkawy|arXiv (Cornell University)|Feb 15, 2021

Motor Control and Adaptation8 references4 citations

TL;DR

This paper presents a mathematical analysis of minimum jerk trajectory (MJT) generation for human hand movements, deriving analytical expressions for position and velocity in both straight and curved point-to-point motions. Using MATLAB simulations, it demonstrates that MJT produces the smoothest possible trajectories, minimizing jerk and closely matching human motor behavior.

ABSTRACT

In this chapter, the mathematical analysis of the minimum jerk trajectory (MJT) generation is performed. In this study, the position and the velocity of the minimum jerk trajectory as a function of time is presented in two cases: the first one is the unconstrained point-to-point movements of the human hand, whereas the second case is the curved point-to-point movements of the human hand. Simulation study is carried out with some examples and MATLAB is used for this simulation. The results prove that the minimum jerk trajectory is the smoothest possible movement of the human hand.

Motivation & Objective

• To mathematically analyze minimum jerk trajectory (MJT) generation for unconstrained human hand movements.
• To derive closed-form expressions for position and velocity as functions of time in both straight and curved point-to-point movements.
• To validate the smoothness of MJT through simulation and compare it to human motor behavior.
• To provide a theoretical foundation for MJT in robotics and neuroscience applications.
• To demonstrate that MJT minimizes jerk, thus producing the smoothest possible movement.

Proposed method

• Formulates the minimum jerk problem as a variational calculus optimization, minimizing the integral of the squared third derivative of position (jerk).
• Derives analytical solutions for position and velocity profiles under boundary conditions: initial and final positions and velocities set to zero.
• Applies the solution to both straight-line and curved trajectories using parametric equations.
• Uses time scaling to ensure smooth transitions between start and end points with zero initial and final velocity.
• Employs MATLAB for numerical simulation and visualization of MJT profiles across different movement types.
• Validates the smoothness of MJT by analyzing jerk minimization and trajectory continuity.

Experimental results

Research questions

• RQ1 What are the analytical expressions for position and velocity in minimum jerk trajectories for straight and curved movements?
• RQ2 How does the minimum jerk principle produce the smoothest possible human hand movements?
• RQ3 To what extent does the MJT model replicate observed human motor behavior in point-to-point movements?
• RQ4 What mathematical formulation enables the generation of smooth, continuous trajectories with zero initial and final velocity?
• RQ5 How can the minimum jerk trajectory be generalized to curved paths using parametric representation?

Key findings

• The minimum jerk trajectory produces the smoothest possible movement by minimizing the integral of the squared third derivative of position (jerk).
• Analytical solutions for position and velocity are derived as cubic polynomials in time for straight-line movements with zero initial and final velocity.
• For curved movements, the trajectory is parameterized using a time-varying path, and the minimum jerk solution is obtained by optimizing the path parameterization.
• MATLAB simulations confirm that the MJT exhibits minimal jerk and high smoothness, with continuous acceleration and zero initial and final velocity.
• The results demonstrate that the minimum jerk model closely matches human motor behavior in point-to-point movements.
• The mathematical formulation provides a robust and generalizable framework for trajectory generation in robotics and biomechanics.