A General Formulation for Path Constrained Time-Optimized Trajectory Planning with Environmental and Object Contacts

Abstract: A typical manipulation task consists of a manipulator equipped with a gripper to grasp and move an object with constraints on the motion of the hand-held object, which may be due to the nature of the task itself or from object-environment contacts. In this paper, we study the problem of computing joint torques and grasping forces for time-optimal motion of an object, while ensuring that the grasp is not lost and any constraints on the motion of the object, either due to dynamics, environment contact, or no-slip requirements, are also satisfied. We present a second-order cone program (SOCP) formulation of the time-optimal trajectory planning problem that considers nonlinear friction cone constraints at the hand-object and object-environment contacts. Since SOCPs are convex optimization problems that can be solved optimally in polynomial time using interior point methods, we can solve the trajectory optimization problem efficiently. We present simulation results on three examples, including a non-prehensile manipulation task, which shows the generality and effectiveness of our approach.

• The paper presents a general Second-Order Cone Programming (SOCP) formulation for path-constrained time-optimal trajectory planning that incorporates environmental and object contacts.
• The methodology uses a convex SOCP approach to integrate dynamics, actuator limits, and nonlinear friction cone constraints for precise contact handling.
• Simulation results demonstrate the method's efficacy and general applicability across different manipulation tasks involving contact, offering practical and theoretical advancements in robotic trajectory planning.

A General Formulation for Path Constrained Time-Optimized Trajectory Planning with Environmental and Object Contacts

The paper "A General Formulation for Path Constrained Time-Optimized Trajectory Planning with Environmental and Object Contacts" addresses significant challenges in the field of robotic manipulation, particularly focusing on achieving time-optimal trajectories for robotic systems that interact with both their environment and held objects. The authors present a comprehensive approach that integrates dynamic constraints, actuator limits, and contact conditions into a Second-Order Cone Programming (SOCP) framework, offering an efficient method to compute time-optimal trajectories for complex manipulation tasks.

Summary

Problem Statement and Challenges

The core problem tackled by the authors is the computation of joint torques and grasp forces for time-optimal manipulation of an object. This involves maintaining a stable grasp on the object and satisfying various constraints related to dynamics, environmental contact, and no-slip conditions at the contact points. The complications arise from the inherent difficulties of accounting for both object-environment and hand-object interactions within the trajectory planning process. Traditional motion planning methods often separate path and time parameterization into distinct steps, which can lead to inefficiencies and suboptimal solutions, particularly when contacts are involved.

Methodology

The authors propose an innovative SOCP formulation that consolidates the path planning and time-optimal control into a single framework. The proposed model acknowledges nonlinear friction cone constraints at both the hand-object and object-environment contacts. Unlike linear or polygonal approximations traditionally used, the authors employ a full nonlinear model for the friction cone, allowing for a more precise representation of the contact dynamics. The key advantage is that SOCP is a convex optimization problem, providing optimal solutions in polynomial time using interior point methods.

Simulation Results

The paper discusses simulation results from three distinct examples, including a non-prehensile manipulation task, demonstrating the efficacy and general applicability of the proposed approach. These simulations highlight the model’s ability to handle varying types of contacts and dynamics constraints, showcasing significant improvements in computational efficiency and adaptability to different task setups.

Practical and Theoretical Implications

The implications of this research are multifaceted. Practically, the development of a robust, efficient method for computing time-optimal trajectories can directly enhance industrial productivity and extend the capabilities of service robots in everyday environments. Theoretically, this work contributes to the understanding of SOCP in robotics, particularly in contexts where contact plays a crucial role.

Future Directions

The research lays the groundwork for several future developments. One potential direction is the extension of this framework to handle dynamic environments where the robot and object interactions are continuously changing. Additionally, integrating real-time adaptive capabilities to account for uncertainty and noise in the system can further improve the robustness of the trajectories computed by this method. Exploring the application of this approach to multiple manipulators and coordinated systems presents another avenue for expanding its utility and effectiveness.

This paper offers a critical advancement in the field of robotic trajectory planning, bridging the gap between theoretical formulations and practical applications, and paving the way for more intelligent and efficient robotic systems capable of complex manipulation tasks in dynamic environments.