[Paper Review] Dynamic Models of Spherical Parallel Robots for Model-Based Control Schemes

Ali Hassani, Abbas Bataleblu+3 more|arXiv (Cornell University)|Oct 1, 2021

Robotic Mechanisms and Dynamics59 references4 citations

TL;DR

This paper presents a systematic derivation of explicit, linear regressor, and Slotine-Li (S-L) regressor forms of dynamic models for spherical parallel robots (SPRs) using the principle of virtual work in task space. The method enables accurate model-based control and parameter identification, validated via ADAMS simulations on ARAS-Diamond and 3-RRR SPRs with closed-form, unique dynamic formulations.

ABSTRACT

In this paper, derivation of different forms of dynamic formulation of spherical parallel robots (SPRs) is investigated. These formulations include the explicit dynamic forms, linear regressor, and Slotine-Li (SL) regressor, which are required for the design and implementation of the vast majority of model-based controllers and dynamic parameters identification schemes. To this end, the implicit dynamic of SPRs is first formulated using the principle of virtual work in task-space, and then by using an extension, their explicit dynamic formulation is derived. The dynamic equation is then analytically reformulated into linear and S-L regression form with respect to the inertial parameters, and by using the Gauss-Jordan procedure, it is reduced to a unique and closed-form structure. Finally, to illustrate the effectiveness of the proposed method, two different SPRs, namely, the ARAS-Diamond, and the 3-RRR, are examined as the case studies. The obtained results are verified by using the MSC-ADAMS software, and are shared to interested audience for public access.

Motivation & Objective

• To develop a unified, analytical framework for deriving multiple dynamic model forms of spherical parallel robots (SPRs) essential for model-based control.
• To overcome the complexity of deriving dynamics for SPRs due to their closed-loop kinematics and inherent kinematic constraints.
• To provide explicit, linear, and S-L regressor forms of SPR dynamics that are suitable for advanced control schemes and parameter identification.
• To ensure the derived formulations are unique and closed-form through Gauss-Jordan reduction, enhancing numerical stability and reproducibility.
• To validate the proposed dynamic models using multi-body dynamics simulation (MSC-ADAMS) on two benchmark SPRs: ARAS-Diamond and 3-RRR.

Proposed method

• Formulate the implicit dynamic model of SPRs using the principle of virtual work in task space, leveraging the robot's kinematic structure.
• Derive the explicit dynamic formulation by transforming the implicit model, separating mass, Coriolis/centrifugal, and gravity terms.
• Reformulate the explicit dynamics into a linear regression form with respect to inertial parameters for use in adaptive control and system identification.
• Further derive the Slotine-Li (S-L) regressor form, which avoids the need for inverse mass matrix computation and acceleration measurements, enhancing robustness.
• Apply the Gauss-Jordan elimination procedure to reduce the dynamic model into a unique, closed-form structure, ensuring consistency and numerical reliability.
• Validate the derived models using MSC-ADAMS simulations on two SPRs: ARAS-Diamond and 3-RRR, using predefined physical and geometric parameters.

Experimental results

Research questions

• RQ1 How can the dynamic model of a spherical parallel robot be systematically derived in explicit, linear, and S-L regressor forms using task-space virtual work?
• RQ2 What is the role of the Gauss-Jordan procedure in achieving a unique and closed-form dynamic formulation for SPRs?
• RQ3 How do the derived dynamic models compare in accuracy and consistency with multi-body dynamics simulation results?
• RQ4 To what extent do the derived formulations support advanced model-based control and parameter identification in SPRs?
• RQ5 Can the proposed method be generalized to other SPR architectures beyond ARAS-Diamond and 3-RRR?

Key findings

• The derived dynamic models for ARAS-Diamond and 3-RRR SPRs show strong agreement with MSC-ADAMS simulation results, confirming the accuracy of the analytical formulations.
• The Gauss-Jordan procedure successfully reduced the dynamic model to a unique, closed-form structure, eliminating ambiguity in parameter estimation.
• The explicit dynamic formulation enables precise separation of inertial, Coriolis/centrifugal, and gravity effects, essential for inverse dynamics control.
• The linear and S-L regressor forms are validated as suitable for adaptive control and dynamic parameter identification, with the S-L form offering computational advantages.
• The method provides a scalable and reproducible framework for deriving dynamic models of SPRs, applicable to various robot architectures.
• The inclusion of detailed physical parameters (mass, center of mass, inertia) in Tables 3 and 4 enables direct use in control and simulation environments.