Minimum Jerk Trajectory Generation for Straight and Curved Movements: Mathematical Analysis

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.

• The paper introduces a mathematical framework for generating minimum jerk trajectories in both straight and curved point-to-point movements.
• It employs fifth-order polynomial solutions and nonlinear constraint resolution to ensure smooth initiation and termination of movements.
• Simulations validate the approach for robotic control, human-robot interaction, and rehabilitation, confirming unique and optimal trajectory solutions.

Minimum Jerk Trajectory Generation for Straight and Curved Movements: Mathematical Analysis

Overview

This work presents a rigorous mathematical treatment of minimum jerk trajectory (MJT) generation for straight and curved point-to-point (PTP) movements, emphasizing both unconstrained and constrained hand path scenarios. The chapter systematically derives solutions for minimum jerk positioning and velocity, under both collinear and intermediary waypoint constraints. Simulations illustrate the theoretical constructs, and the applicability to advanced domains such as human-robot interaction, robotic control, and rehabilitation is highlighted.

Minimum Jerk Trajectories: Mathematical Derivation

The minimum jerk trajectory paradigm arises from variational optimization, targeting the minimization of jerk, the third time derivative of position, over a specified time interval. For unconstrained, straight-line motion, the boundary conditions assume zero initial and terminal velocity, leading to a fifth-order polynomial position profile, parameterized by normalized time. The canonical form is:

$x(t) = x_0 + (x_f - x_0)(6t^5 - 15t^4 + 10t^3),$

where $t = \tau / t_f$, with $\tau$ as the current time and $t_f$ the total movement time. The corresponding velocity profile, obtained by differentiation, yields:

$$\dot{x}(t) = (x_f - x_0)(30t^4 - 60t^3 + 30t^2) / t_f.$$

This form enforces smooth initiation and termination, characteristic of coordinated biological movement and desirable in high-dexterity robotics.

For spatially constrained, curved PTP motion, the trajectory must interpolate an intermediate via-point at a free, but internally consistent, time parameter $t_1$. The position and velocity profiles are partitioned piecewise at $t_1$ and must satisfy both endpoint and via-point constraints, resulting in a high-order polynomial system. The core equality constraints and the determination of the auxiliary time parameter ($t_1$) are cast as nonlinear equations, solvable through symbolic and numeric computation.

Simulation Studies and Numerical Insights

Multiple simulation scenarios are presented:

• Pure axes motion: The straight-line case validates the theoretical minimum jerk solution, with trajectories exhibiting zero velocity at endpoints and a smooth, unimodal velocity profile.
• Curved constrained motion: By introducing waypoints, the trajectory demonstrates continuous, smooth paths that deviate from a single axis. The via-point time $t_1$ is found analytically via root-finding on the derived polynomial constraints.

Four representative cases with varying boundary and intermediary conditions are explored. The position and velocity plots from these simulations confirm the preservation of jerk minimization even under nontrivial waypoint constraints. In each case, the single viable solution for the waypoint passage time lies within the unit interval, signifying the physical plausibility and uniqueness of the solution.

Application Domains

The theoretical development is anchored by several explicit application domains:

• Variable Admittance Control: MJTs serve as reference profiles for online adaptation of robot admittance parameters via neural networks. The error between the MJT reference and actual velocity modulates virtual inertia/damping for compliant human-robot interaction.
• Rehabilitation and Haptics: In motor retraining and assistive mechatronics, MJTs guide patient actuations to suppress unintentional oscillatory behaviors (tremor, hesitation) and ensure comfort and efficacy.
• Mobile Robotics: For nonholonomic platforms, MJT-based planning constrains paths to minimize actuator-induced vibrations and mechanical wear, subject to spatial and temporal waypoints.

In power reactor control, jerk minimization of reference signals has been demonstrated to prolong actuator life and smooth dynamic responses.

Theoretical and Practical Implications

From a theoretical standpoint, this analysis affirms that MJT-based frameworks are globally optimal for biological and artificial movement between PTP locations, even in the presence of intermediate via-point constraints. The analytic tractability, especially for curved motions, enables systematic deployment in complex task spaces.

Practically, the explicit derivation and accompanying computational procedures (as exemplified by the MATLAB code) streamline integration into advanced robotic control architectures and clinical/prosthetic devices, where real-time smoothness and adaptability are paramount.

Future Directions

Future investigations may extend this framework to:

• High-dimensional configuration spaces, for articulated manipulators or redundant systems.
• Real-time adaptation under model uncertainty and unstructured environments.
• Integration of MJT with learning-based control schemes for hybrid optimality and adaptability.
• Exploration of the biological plausibility of MJT deviations under pathological conditions.

Conclusion

This chapter establishes a complete mathematical framework for MJT generation in both straight and curved PTP human and robotic movement, supported by numerical simulations and robust application scenarios. The formalism captures the essential features of smooth motor coordination and underpins methodologies in human-robot interaction, rehabilitation, haptics, and optimal robotic motion planning (2102.07459).