Research paper
Kinematics, dynamics, and load distribution analysis of a 4-PPPS redundantly actuated parallel manipulator

https://doi.org/10.1016/j.mechmachtheory.2021.104494Get rights and content

Highlights

  • Kinematics and dynamics analysis with the screw theory.

  • Determination of the major joint set for the kinematics and dynamics analysis.

  • Solve the joint force outputs for non-redundant and fully redundant mode.

  • Load distribution algorithm for avoiding joint overload and improving precision.

Abstract

In this paper, the kinematics, dynamics, and load distribution algorithm of a 4-PPPS redundantly actuated parallel manipulator are investigated with the screw theory. For the forward kinematic analysis, we propose the major joint set, a minimum set for determining the kinematic and dynamic state of the mobile platform with six specially selected prismatic joints. The reciprocity of screws is applied for simplifying the derivation of the Jacobian matrix. The driving forces of the active joints are solved with the principle of virtual work for the non-redundant and the fully redundant actuation mode with the major joint set. For the fully redundant actuation mode, to avoid joint overload resulted from the conventional least-norm load distribution solution, to smooth the force transition, and to increase the positioning accuracy of the manipulator, a load distribution algorithm based on Lagrange optimization is proposed. These effects are achieved by means including reducing the deformations of the z-axis links and considering the sensitivity and elasticity of major joints. At last, the effectiveness of the modeling method and the proposed algorithm is verified with a case study.

Introduction

In the last few decades, redundant parallel manipulators (RPMs) have gained increased attention from robot researchers due to better stiffness, greater payload capacity, and higher agility than their serial counterparts. The mobility of an RPM is the minimum number of independent joint variables to fully determine the end-effector’s motion. The degree-of-freedom (DoF) is defined as the minimum number of variables needed to define the end-effector’s motion relative to a fixed reference frame. When the mobility of an RPM is greater than its DoF, the RPM is called a kinematically redundant parallel manipulator (KRPM). When the number of actuators in the RPM is greater than the mobility, the RPM is called a redundantly actuated parallel manipulator (RAPM) [1]. Generally, redundancy enhances the performance of RPMs in kinematics, including singularity elimination, workspace enlargement, and dexterity enhancement. While the benefits of redundancy in dynamics include force transmission optimization, impact compensation, fault tolerance improvement, and load distribution adjustment [2].

Among different RAPMs, the n-PPPS RAPMs are widely employed for large parts assembly in the aviation manufacturing industry. Compared to conventional PMs such as Gough–Stewart platforms (GSP), n-PPPS RAPM can decouple its motion into different joints. A n-PPPS RAPM usually possesses a regular-shaped workspace. This property means simpler boundary conditions for motion planning, free of identifying [3], [4], [5] or designing [6] regular-shaped subspaces within the workspace, which is usually an intricate task for platforms like GSP and Orthoglide. More specifically, for a 4-PPPS RAPM with a rectangle mobile platform, the reachable workspace and constant orientation workspace are cuboids. The 4-PPPS RAPM does not have singularity inside its reachable workspace [7]. While for GSPs, their singularities will easily cause self-motion, which is unwanted in the assembly industry. The n-PPPS RAPM is easier to reconfigure compared to other RAPMs like ART [8]. In the aviation industry, typical applications include 3-PPPS RAPMs from [9], [10], [11], [12]. However, 3-PPPS RAPM can have unbalanced loads. 4-PPPS RAPM [13] or n-PPPS RAPM with n>3 can be used to overcome this issue.

Many research works have studied n-PPPS RAPMs. Work by Zhang et al. [10] establishes the kinematic equations of a 3-PPPS RAPM and uses the Newton–Euler algorithm for the dynamics of the alignment system. The pseudo-inverse method of the full Jacobian matrix is used for the redundant driving forces. This method is straightforward, but it cannot deal with situations when joint overload occurs under heavy loads. Work by Lochte et al. [14] compares the position accuracy of a 3-PPPS tripod structure against a 6-DoF 6-SPS hexapod structure and, in the meantime, proposed an optimized configuration of a 3-PPPS RAPM. Though they argue that error transmission of the tripod structure is greater than the hexapod structure, they do not consider the error reduction resulting from the redundancy of tripods. To reduce this positioning error, work by Qi et al. [15] puts forward a kinematic error calibration method for a dual 4-PPPS RAPM aircraft docking system. Nevertheless, Qi et al. do not analyze the dynamic errors caused by the deformation of the manipulator or the load. A control system for a 3-PPPS RAPM is designed in [16], in which a group of six joints is position-controlled, and the remaining joints are force-controlled to reduce the deformation of the load. However, the choice of these six joints is not systematically analyzed. If the force-controlled joints can share loads of the position-controlled joints, the dynamic redundancy of this manipulator would be better exploited.

Though the manipulator dynamics can be solved with the Newton–Euler method or Lagrange method, these methods need to model the manipulator with differential equations. Kane’s method improves the Lagrange method by introducing partial velocity and partial angular velocity to set up the equality between the generalized active forces and generalized inertia forces, yet still uses differential equations. For a highly redundant manipulator with closed kinematic chains, building and solving these equations are computationally exhausting. The screw theory can be used as an alternative tool in parallel manipulator modeling. It simplifies the computation of dynamics to matrix multiplication, which is suitable for a modern computer. Previous works in [17], [18], [19] are efficient in deriving the kinematics and dynamics input–output relationship of other manipulators. However, owing to the high redundancy of the n-PPPS RAPMs and difficulty in measuring the joint velocities of the spherical joints, existing modeling methods with screw theory cannot be directly applied.

The n-PPPS RAPMs can operate under different modes. The most frequently used ones are the non-redundant and fully redundant actuation mode. For non-redundant actuation mode, the solution to the joint output forces is unique since only the major joints have active force outputs. While for fully redundant actuation mode, all the joints are active. There will be infinite solutions for the joint output forces due to the actuation redundancy, endowing the manipulator with load distribution potential. From the standpoint of close loop manipulator performance improvement [20], shocking reduction [21], fault tolerance [22] and joint overload avoidance [23], previous studies have presented some feasible methods optimizing the force distribution of manipulators with low mobility. A weighted pseudo-inverse algorithm is used in [24], and a quasi-Newton algorithm is used in [25]. They both reduce the maximum joint force of the RAPMs, but no joint force limit is considered. In [26] and [27], joint force limit is taken into consideration with singular value decomposition-based algorithm [26] and Lagrange algorithm [27]. However, none of the algorithms tries to optimize the load distribution for positioning accuracy improvement, which could be achieved by distributing the load for less deformation on the links. In this case, the positioning accuracy is defined as the mean of positioning error for the end-effector. This performance improvement is significant for accurate robotic tasks, such as assembly and machining.

Therefore, in this paper, to concisely develop a model methodology of the kinematics and dynamics of n-PPPS RAPM, the screw theory is systematically applied to a 4-PPPS RAPM. For the forward kinematics, a new concept named the major joint set is introduced as a set of six prismatic joints to determine the kinematic state of the mobile platform. The reciprocity of screws is utilized to derive the Jacobian matrix. With the help of the major joint set, the joint output forces of the active joints are determined for the non-redundant actuation mode and fully redundant actuation mode. To further utilize the redundancy of the manipulator, a load distribution algorithm based on Lagrange optimization is proposed to improve the accuracy of the manipulator by reducing the deformation of the links and avoiding joint overload under the fully redundant actuation mode. In the end, the derived kinematic model, dynamic model, and the proposed load distribution algorithm are applied to a case study of a 4-PPPS RAPM. The simulation results show distinct load capacity improvement and positioning error reduction by load distribution.

Section snippets

Model description

The schematic of the 4-PPPS RAPM is shown in Fig. 1. The RAPM comprises four limbs connecting a rectangular mobile platform of size 2lx×2ly. The bodies are numbered from the bottom to the top, where the fixed base is denoted as body 0, and the mobile platform is denoted as body 6. A body fixed point op is located at the center of the rectangle with a reference frame P attached to it. Each limb contains three mutually orthogonal prismatic joints Pi1, Pi2, and Pi3, which connect the first link,

Kinematics

Based on the screw system introduced in the previous section, the kinematic chain of the 4-PPPS RAPM can be obtained using the screw theory. The inverse kinematics is firstly derived with the definition of the reciprocal screws. Then the forward kinematics of the manipulator is derived with a newly introduced concept as the major joint set. The selection principle of these joints is also discussed.

Dynamics

In this section, the joint output forces are analyzed based on the principle of virtual work. Usually, it is hard to obtain the equation of joint output forces of this redundant manipulator with six redundant joints. However, when the major joint set is applied, the equation of joint forces can be obtained for the non-redundant actuation mode and fully redundant actuation mode. The operating mode depends on the number of active joints. When only the major joints are active, the manipulator

Case study

To demonstrate the kinematics, dynamics and load distribution method of the 4-PPPS RAPM implemented in this paper, a numerical case study is provided. In this case, the mobile platform of the 4-PPPS RAPM in Fig. 4 is a square given the size lx=ly=0.25 m. All the prismatic joints have the same moving range as lmax=0.1 m and lmin=0.1 m. The initial configuration of the manipulator is that all the joint displacement are 0, the center of spherical joints expressed in B are S1=0.25,0.25,0 m, S2=0.

Conclusion

In this paper, the kinematics and dynamics of a 4-PPPS RAPM are solved by the theory of screw and the principle of virtual work. To solve the forward kinematics of this highly redundant parallel manipulator, a new concept named the major joint set is introduced to minimally determine the kinematic state of the manipulator. The selection method of the major joints is deducted in this paper by analyzing the Jacobian matrix formed by the reciprocal screws of the prismatic joints. This analysis can

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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    Research supported by National Natural Science Foundation of China under Grant No. U1813208 and National Key Research and Development Program of China under Grant No. 2019YFB1312703.

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