Freedom Analysis and Kinematic Study of a Double-Layer 3-RSR Parallel Mechanism

Freedom Analysis and Kinematic Study of a Double-Layer 3-RSR Parallel Mechanism

Xinyi Yang

(Corresponding author) School of Mechanical Engineering Tianjin University of Technology and Education Tianjin 300222China

Yuhao Yang

School of Mechanical Engineering Tianjin University of Technology and Education Tianjin 300222China

Rui Zhang

School of Mechanical Engineering Tianjin University of Technology and Education Tianjin 300222China

Yuhan Ao

School of Mechanical Engineering Tianjin University of Technology and Education Tianjin 300222China

Minjie Cheng

School of Mechanical Engineering Tianjin University of Technology and Education Tianjin 300222China

Zisun Huang

School of Mechanical Engineering Tianjin University of Technology and Education Tianjin 300222China

Abstract – To address the limitations of traditional single-layer parallel mechanisms in terms of restricted working space and insufficient reachability, a double-layer 3-RSR parallel mechanism is proposed. This study focuses on the degree-of- freedom analysis and kinematic modeling of this mechanism. Based on screw theory, the degrees of freedom of the double-layer 3-RSR parallel mechanism are systematically analyzed, confirming it possesses three degrees of freedom: two rotational degrees of freedom around the x-axis and y-axis, and one translational degree of freedom along the z-axis. By establishing fixed and moving coordinate systems, the positive kinematic models for both single-layer and double-layer 3-RSR mechanisms were derived, yielding position and orientation parameters for the moving platform. The validity of the kinematic models was verified through joint simulation using MATLAB and SolidWorks. Results demonstrate substantial agreement between theoretical and simulated trajectories, confirming the mechanism’s capability for precise control of the moving platform’s path. This provides a theoretical foundation for the engineering application of double-layer 3-RSR parallel mechanisms.

Keywordsparallel mechanism; 3-RSR; degrees of freedom; kinematic analysis; screw theory

1. INTRODUCTION

   With the advancement of new industrialization and the digital transformation of traditional industries, the manufacturing sector demands higher levels of automation and precision in equipment (1). Parallel mechanisms, leveraging advantages such as high stiffness, high precision, and compact structure, have found extensive applications in precision positioning, industrial robots, and other fields (2). Compared to serial mechanisms, parallel mechanisms offer superior load- bearing capacity and motion stability, demonstrating significant application potential in complex operating conditions (3). Among these, the 3-RSR parallel mechanism, as a typical low-

   Supported by the National College Students’ Innovation and Entrepreneurship Training ProgramProject ID: 202510066034

   degree-of-freedom parallel mechanism, features a simple structure and convenient control, making it commonly used in scenarios requiring two rotary and one translational (2R1T) motions (4). However, single-layer 3-RSR mechanisms suffer from structural limitations, including restricted working space and inadequate reachability, making them ill-suited for complex tasks (5). Consequently, innovatively designing double-layer 3- RSR parallel mechanisms through stacked layering has emerged as an effective approach to expand working space and enhance reachability (6).

   Degree-of-freedom analysis forms the foundation for parallel mechanism design and application, directly determining the mechanism’s motion characteristics and operational range

   (7). Spiral theory serves as an effective tool for analyzing parallel mechanism degrees of freedom, enabling precise description of rigid body motion and constraint relationships between components (8).Numerous scholars have conducted in- depth studies on parallel mechanism degree-of-freedom analysis based on spiral theory: Wei et al. proposed a comprehensive method for non-coupled 2R1T parallel mechanisms based on branch driving force spiral theory, providing a theoretical basis for low-degree-of-freedom parallel mechanism design (9); Sun et al. employed the finite motion spiral method to describe and analyze the finite motion characteristics of 1T2R parallel mechanisms with parasitic motion (10);Fang et al. designed a 2R1T redundant-drive parallel mechanism with enhanced motion flexibility and analyzed its degrees of freedom based on spiral theory (11). These studies have advanced the development of degree-of-freedom analysis techniques for low-degree-of- freedom parallel mechanisms. However, research on degree-of- freedom analysis for double-layer stacked 3-RSR parallel mechanisms remains scarce, and the relevant theoretical framework is still incomplete (12).

   Kinematics analysis constitutes a vital component of parallel mechanism research, involving the determination of position, velocity, and acceleration to provide a foundation for mechanism control and performance optimization (13). Regarding the kinematics analysis of 3-RSR parallel

   mechanisms, scholars worldwide have achieved a series of results: Wang et al. conducted inverse kinematics and workspace analysis on the 2-RPU/UPRS mechanism, verifying its smooth motion characteristics under large-angle conditions through Adams simulation (14);Li et al. added angle sensors at the ball joints of the 3-RSR mechanism and constructed constraint equations to resolve the multiple solutions problem in forward kinematics (15); Herrero et al. optimized the 2PRU- 1PRS parallel robot based on workspace and power consumption criteria and conducted in-depth kinematic analysis (16).However, the double-layer 3-RSR parallel mechanism exhibits more complex kinematic characteristics than single-

   kinematic analysis, particularly in solving the motion of multi- degree-of-freedom parallel mechanisms.

   If a rigid body undergoes two consecutive finite displacements,,1 and,2 , these can be represented by finite rotors respectively.

   layer mechanisms due to the superposition of upper and lower

   layers, rendering existing single-layer kinematic analysis methods inapplicable (17). Therefore, targeted research on

   kinematic modeling and solution for double-layer 3-RSR parallel mechanisms holds significant theoretical and engineering importance (18).

   In summary, this study focuses on the degree-of-freedom analysis and kinematics research of double-layer 3-RSR parallel

   The final continuous motion of the rigid body is then,12:

   mechanisms. Based on screw theory, the mechanism’s degrees

   of freedom are analyzed to clarify its motion characteristics. Positive kinematic models for both single-layer and double- layer mechanisms are established, with model validity verified through simulation. This provides theoretical support for the

   mechanism’s design and application, holding significant engineering value for advancing the expanded application of low-degree-of-freedom parallel mechanisms (19-25).

   where the symbol “” denotes the trigonometric product

   operation of rotors, and “,2× ,1 ” denotes the multiplication operation of rotors.

2. SPIRAL THEORY FUNDAMENTALS

   Rigid-body motion is typically categorized into finite motion and instantaneous motion. Finite rotors describe continuous

   motion processes, while instantaneous rotors characterize instantaneous motion states. Compared to traditional

   mathematical tools, FIS theory uniformly expresses rigid-body motion in a concise six-dimensional pseudo-vector form, achieving the organic integration of finite and instantaneous

   rotors through differential mapping.

   According to Chasles’ theorem, any rigid body motion between two poses can be equivalently represented as a helical motion around a fixed spatial axisa composite motion involving rotation about the axis and translation along it. Thus,

   finite rigid body motion can be fully characterized by the spatial

   The finite motion of a rigid body can be described as the

   cumulative result of multiple rotations and translations. In finite rotors, the finite motion of the nth motion joint in the nth branch can be expressed as

   position , rotation angle , and translation distance of this

   axis. The fundamental expression for finite rotors is:

   In this expression, the finite rotor represents the process of a rigid body undergoing a pose transformation described by : rotating about the axis with angular displacement and translating along the axis direction . Here, denotes the unit direction vector of the motion axis, while corresponds to the position vector of the axis. The calculation of finite rotors is based on rotor trigonometric operations, which are inherently complex nonlinear computations. They characterize continuous rigid-body motion by integrating rotation and translation parameters to achieve precise representation of spatial pose evolution. Such operations play a pivotal role in rigid-body

3. DEGREE OF FREEDOM ANALYSIS OF A DOUBLE-LAYER 3-RSR PARALLEL MECHANISM
   1. Structure of the Double-Layer 3-RSR Parallel Mechanism

      The double-layer 3-RSR parallel mechanism consists of two single-layer 3-RSR parallel mechanisms stacked vertically. It comprises a fixed platform, an intermediate platform, a moving platform, and three symmetrically distributed RSR branches. Each branch connects the fixed platform, intermediate platform, and moving platform through rotary pairs (R) and spherical joints (S) and rotary joints (R) to sequentially connect the fixed platform, intermediate platform, and moving platform. Drive joints are arranged on the fixed platform side, enabling attitude adjustment and positional movement of the moving platform

      through branch chain transmission. The mechanism employs a layered stacking design. The upper and lower 3-RSR mechanisms achieve motion linkage through the intermediate platform. This design retains the high-precision motion characteristics of a single-stage 3-RSR mechanism while enhancing the spatial reachability of the end-effector through

      hierarchical coordination. Its structural properties align closely with the design requirements of the 1T2R parallel mechanism.

   2. stablishment of the Screw System

      Dynamic Platform

      Branch 1

      Branch 3

      Branch 2

      Fixed platform

      Branch 2 Branch 3

      Branch 1

      Fig. 1. Parallel Topology Mechanism

      To analyze the degrees of freedom of the double-layer 3-RSR parallel mechanism, a branch coordinate system is first established. The origin of the branch coordinate system is selected as the center of the intermediate platform’s pivot pair, where the z-axis is perpendicular to the plane of the intermediate platform and points toward the moving platform, the positive x- axis direction points toward the geometric center of the intermediate platform, and the y-axis direction is determined by the right-hand rule. Based on the mechanism’s symmetry, the center of rotation (111 ) lies on the axis of the branch

      chain ( ), and the center of rotation (11 ) exhibits central

      symmetry about the origin. Consequently, the coordinate parameters of the center of rotation (1(0 0 ) ) and the center of rotation ( 1(0 0 )1( 0 )

      1( 0 ) ) can be derived. According to the spatial

      orientation of each motion spiral in the figure, the expression for the motion spiral system of this branch chain can be constructed. The expression is:

   Analysis of the RSR branch’s kinematic spiral system reveals a rank of 6, indicating the branch imposes no constraints on the platform. Since the three branches are symmetrically distributed and structurally identical, the kinematic spiral system of the entire double-layer 3-RSR parallel mechanism is the intersection of the three branch systems. Thus, the moving platform possesses 6 degrees of freedom. For a single-layer 3-RSR parallel mechanism, when input conditions are specified, the motion state of the moving platform is deterministic. The moving platform, possessing 2R1T degrees of freedom, can achieve rotation and translation within a defined spatial range. For the double-layer 3-RSR parallel mechanism, the intermediate platform’s link functions solely as a single-degree- of-freedom transmission component, introducing no additional degrees of freedom. Therefore, the total degrees of freedom for a double-layer 3-RSR parallel mechanism are 3: rotation about the x-axis, rotation about the y-axis, and translation along the z- axis. This degree-of-freedom configuration satisfies the “1 translation + 2 rotations” motion requirements for most precision positioning scenarios, ensuring flexible pose adjustment of the moving platform within its workspace.

   Fig. 2. Branch Coordinate System of Double-Layer 3-RSR Parallel Mechanism

4. KINEMATIC ANALYSIS OF THE DOUBLE-

   Further, the vectors and are obtained as:

   LAYER 3-RSR PARALLEL MECHANISM 1

   To perform kinematic analysis of the double-layer 3-RSR

   parallel mechanism, one must first complete the kinematic

   analysis of the single-layer 3-RSR mechanism and then extend 2 2

   it to he double-layer mechanism to determine the forward kinematic solution of the double-layer mechanism.

   1. Kinematic Analysis of Single-Layer 3-RSR Mechanism

      Performing the cross product operation on these two vectors yields the normal vector of plane123 , pointing from the stationary platform toward the moving platform. Using the coordinates of point1 and the normal vector, we can derive the point-normal equation of plane 123 . This allows us to calculate the distance from point to plane123 .

      The correct parameters are then:

      Fig. 3. Schematic Diagram of the 3-RSR Mechanism

      As shown in Figure 3, establish the fixed coordinate

      ( 0, 0)

      system and the moving coordinate system1 111

      on the fixed platform and moving platform, respectively. The

      coordinate origins1 are the centers of the fixed and moving

      platforms, respectively. The axis aligns with the vector , while the axis is perpendicular to the fixed platform and points vertically upward. The 1 axis follows the direction of vector , 1 The axis is perpendicular to the moving

      platform. Define as the angle between the projection of the axis onto the base plane and the positive direction of the axis in the fixed coordinate system , as the angle between the moving coordinate system and the axis in the fixed coordinate system

      , as the distance between the centers of the moving and fixed platforms, as the angle between the axis and the z-axis of the fixed coordinate system = , the distance from the platform

      center to the center of the rotating pair is , and the total length

      of all links is .

      Point 123 can be expressed using an equation containing the input angle123 .1

   2. Kinematic Analysis of Double-Layer 3-RSR Mechanism

      Fig. 4. Double-Layer Parallel Mechanism

      The double-layer 3-RSR parallel mechanism can be regarded as a superposition structure of two single-layer 3-RSR parallel mechanisms. When the input angle of the lower mechanism

      takes a specific value, the output angle of the corresponding branch of the upper mechanism will match it. If the input angle changes, the output angle of the corresponding branch of the upper mechanism will also change according to dynamic laws. This relationship determines the motion transmission characteristics of the double-layer parallel mechanism.

      As shown in Figure 4, this mechanism primarily consists of a fixed platform, a moving platform, and support links. The support links are structurally distributed uniformly in a 360- degree circular pattern. The active link connects to the moving platform at one end, responsible for transmitting motion or force inputs. The passive link connects to the stationary platform, serving to stabilize and support the structure. Through this design, the motion of the moving platform can be transmitted to the passive link via the active link, enabling precise motion

      Performing the cross product operation on these two vectors yields the normal vector of plane123 , pointing from the stationary platform toward the moving platform. Using the coordinates of point1 and this normal vector, we can derive the point-normal equation for plane 123 , enabling the calculation of the distance from point to plane123 .

      The solution parameters are:

      control of the mechanism.

      2 222 A fixed coordinate system is established relative to the stationary platform, while a moving

      coordinate system 1 111 is established relative to the moving platform. The coordinate origin12 represents the center point of the stationary and moving platforms,

      respectively. The axis aligns with the direction of vector1 ,

      with the axis perpendicular to the stationary platform and oriented vertically upward. The1 axis aligns with the direction of vector11 , with the1 axis perpendicular to the moving

      platform. The2 axis aligns with the direction of vector22 ,

      with the2 axis perpendicular to the moving platform. Define as the angle between the projection of the2 axis onto the base plane and the positive direction of the axis in the moving

      1 The rotation matrix relative to the system can be expressed as:

      coordinate system, (m, m) as the angle between the moving

      coordinate system and the 2 axis in the fixed coordinate

      system,1 as the center distance between the moving and fixed platforms, as the distance from the platform center to the center of the rotating pair, and as the total length of all linkages. The position of point 123 can be expressed using an

      equation containing the input angle123 .

      where2 is the angle between the system and the fixed coordinate system axes =

      Further, vectors and

      are obtained as: 2

      Further derive the vector:

      2 3 As shown in the figure 6, the theoretical trajectory calculated

      by MATLAB is compared with the simulated trajectory

      3 1

      1 The distance to the plane123 :

      (2( ) + 2 )

      obtained from SolidWorks. The figure indicates that the two trajectories are essentially consistent, with a maxmum positional error of less than 0.1 mm, verifying the correctness of

      the double-layer 3-RSR parallel mechanism’s kinematic model. The simulation results demonstrate that this mechanism can achieve precise control of the moving platform’s trajectory,

      Therefore, the distance and solution parameters for12 are:

      meeting the requirements for precision positioning applications.

      where2, 2, 2 are all functions of and can be indirectly converted into functions of .

      The vector1 can be expressed as:

      Fig. 5. Motion Trajectory of the Double-Layer Parallel Mechanism

      The vector 12 can be expressed in the 1 coordinate system as:

      The vector2 can be expressed as:

      Center distance between moving and fixed platforms:

5. Numerical Examples and Simulation Verification

   The theoretical correctness of the designed double-layer 3- RSR parallel mechanism was verified using MATLAB and SolidWorks software.

   First, the mechanism’s forward kinematics were simulated and verified, with measured link lengths = 0.07848 , and platform radius = 0.04497 . The motion trajectory follows equations (41) to (43), as shown in Figure 5:

   Fig. 6. Simulation of the motion position of a double-layer parallel

6. CONCLUSIONS

This study focuses on the degree-of-freedom analysis and kinematics research of the double-layer 3-RSR parallel mechanism. The main findings are as follows:

1. 1. Based on screw theory, a motion screw system for the RSR branch of the double-layer 3-RSR parallel mechanism was established. Analysis of the mechanism’s degrees of freedom confirmed it possesses three degrees of freedom: two rotational degrees of freedom about the x-axis and y-axis, and one translational degree of freedom along the z-axis. This configuration satisfies the “1 translation + 2 rotations” motion requirement.
   2. By establishing fixed, intermediate, and moving coordinate systems, the positive kinematic models for both single-layer and double-layer 3-RSR mechanisms were derived, yielding position and attitude parameters for the moving platform. The double-layer mechanism’s kinematic model was constructed by superimposing the kinematic models of its upper and lower layers, accurately describing the mechanism’s motion characteristics.
   3. The correctness of the kinematic models was verified through joint MATLAB and SolidWorks simulations. Theoretical trajectories closely matched simulated trajectories, with maximum positional errors below 0.1 mm. This demonstrates the mechanism’s capability for precise motion platform trajectory control, providing a theoretical foundation for its engineering applications.

Future research will focus on inverse kinematics analysis and control strategy development for the double-layer 3-RSR parallel mechanism. Experimental validation using a physical prototype will further enhance mechanism performance and advance its engineering implementation.

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