Session: ASME Undergraduate Student Design Expo

Paper Number: 175772

Digital Finite Time Control of Robotic Arm by Implicit Discretization 

In dynamical systems theory, asymptotic and exponential control are designed to regulate system trajectories toward a desired equilibrium over an infinite time horizon. These approaches are suitable in many settings but are insufficient for applications requiring rapid convergence and high precision, such as robotic manipulation, aerospace guidance, or autonomous swarm coordination. In such cases, the control objective must be achieved within a finite time window, and residual errors must be eliminated to ensure reliability. Finite-time control (FTC) directly addresses this need by enforcing convergence to the target state in a finite time window with minimal error. Building on Lyapunov-based methods, finite-time stability analysis provides guarantees of convergence and robustness, ensuring that the system reaches the desired state without steady-state error. Sliding mode control (SMC) achieves finite-time convergence by enforcing trajectories to evolve along a designed sliding manifold, offering robustness to modeling uncertainties and external disturbances. 

Despite its advantages, practical implementation of FTC poses challenges, particularly when transitioning from continuous-time formulations to discrete-time systems, as required for digital controllers. One of the major challenges is chattering, which manifests as rapid high-frequency oscillations around the equilibrium point. Chattering not only prevents smooth convergence but can also excite unmodeled dynamics, induce actuator wear, and degrade overall performance. This issue is more pronounced in discrete-time implementations, where sampling and quantization effects exacerbate the oscillations. To address this, various strategies have been proposed, including adaptive gain tuning, error bounding techniques, and higher-order sliding mode controllers. While effective to some extent, these methods often introduce trade-offs between convergence speed, implementation complexity, and robustness. 

Implicit discretization has emerged as a promising solution for mitigating chattering in discrete-time FTC. Unlike explicit discretization schemes, which may lose critical dynamical information during sampling, implicit methods preserve the underlying finite-time convergence properties of the continuous-time design. By incorporating future-state dependencies through implicit formulations, these methods provide smoother and more stable convergence trajectories. However, they come at the cost of higher analytical complexity and increased computational requirements, making their practical implementation nontrivial. Recent developments in set-valued analysis and implicit numerical schemes have provided tools to overcome some of these challenges, enabling more rigorous treatment of discrete-time sliding mode dynamics. 

Our preliminary research explores the application of implicit discretization to terminal sliding mode control (TSMC). Specifically, we employ set-valued mapping-based implicit discretization to ensure stability and robustness in discrete-time FTC implementations. In this poster, we apply this framework to the control of a three-degree-of-freedom robotic arm, which serves as a representative system for high-precision motion control under nonlinear dynamics. Simulation studies conducted in the MATLAB environment demonstrate the effectiveness of the proposed method, highlighting smoother convergence and reduced chattering compared to existing FTC approaches. These results illustrate the potential of implicit discretization in bridging the gap between continuous-time theory and discrete-time implementation, paving the way for more reliable and efficient finite-time control in practical robotic systems. 

Presenting Author: Hanna Lutz University of South Carolina

Presenting Author Biography: Hanna Lutz is a senior Aerospace Engineering major with a German minor at the University of South Carolina. She has been involved in undergraduate research with Dr. Lee since the spring of sophomore year. Her interest in controls began due to her love of utilizing mathematics to solve engineering problems. She also has a particular interest in controls due to its relevance to satellites and other aerospace applications. Additionally, she has been a Peer Tutor, Honors Peer Mentor, and was the professional development chair of Alpha Omega Epsilon at the University of South Carolina.

Authors:

Hanna Lutz University of South Carolina
Junsoo Lee University of South Carolina