# Modeling Parallel Robot Kinematics for 3T2R and 3T3R Tasks Using Reciprocal Sets of Euler Angles

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Inverse Kinematics and Resolution of Task Redundancy for Serial-Link Robots

#### 1.2. Overview of Parallel Robots Structures for 3T2R Tasks

- I
- mechanisms with full platform mobility (3T3R) that are redundantly controlled to five DoF,
- II
- mechanisms with 3T2R platform mobility enforced with a passive five-DoF constraining leg and five other legs with six DoF each,
- III
- mechanisms with 3T2R platform mobility resulting from the mobility of five actuated legs with five or six DoF each,
- IV
- mechanisms with 3T2R platform mobility and five legs with only five DoF each.

#### 1.3. Inverse Kinematics of Parallel Robots for 3T2R Tasks

#### 1.4. Motivation and Summary of the State of the Art

- a general kinematics model for parallel robots using the concept of reciprocal Euler angles [5],
- a complete elimination of the redundant operational space coordinate in this formulation for 3T2R tasks allowing a nullspace optimization in the gradient-based inverse kinematics,
- proofs, examples and simulations to show the performance for single serial kinematic leg chains and complete parallel robots.

## 2. Inverse Kinematics Problem for Parallel Robots

- The velocity-based theory of linear transformations used by [11] allows determining the mobility of arbitrary parallel robots. The linear transformation is generalized in [12] to accuracy and stiffness modeling by means of screw theory resulting in the “generalized Jacobian”. Both concepts are similar to (10) and do not provide a direct appliance to solve the IKP, since this requires a formulation of the orientation at position level, not velocity level.
- Exploiting the reduction of end-effector coordinates for 3T2R tasks is not possible, since all end-effector coordinates are included in (7).

## 3. Reciprocal Sets of Euler Angles for the Kinematics of a Serial Leg Chain

## 4. Full Kinematic Constraints for Parallel Robots Using Reciprocal Sets of Euler Angles

## 5. Differential Kinematics for Parallel Robots

#### 5.1. Constraint Gradients for the Leading Leg of the 3T2R and All Legs of the 3T3R Case

#### 5.2. Constraint Gradients for the Following Leg in the 3T2R Case

#### 5.3. Gradient-Based Solution of the Inverse Kinematics Problem with Redundancy Resolution

#### 5.4. Differential Kinematics for the Parallel Robot and Its Applications

## 6. Results

#### 6.1. Resolution of Functional Redundancy of a Serial-Link Six-DoF Robot in 3T2R tasks

#### 6.2. Resolution of Functional Redundancy of a Parallel Robot in 3T2R Tasks

#### 6.3. Statistic Results for the Inverse Kinematics of Serial Link Chains

## 7. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

PKM | parallel kinematic machine (parallel robot) |

IKP | inverse kinematics problem |

DoF | degrees of freedom |

xTyR | x translational and y rotational degrees of freedom |

## Appendix A. Mathematical Symbols for Reciprocal Euler Angles in Inverse Kinematics

#### Appendix A.1. Proof for the Properties of Reciprocal Euler Angles

#### Appendix A.2. Relation of the Geometric Jacobian and the Partial Derivative of the Rotation Matrix

#### Appendix A.3. Relation of the Inverse- and Direct-Kinematics Matrices to the Analytic Jacobian

#### Appendix A.4. Matrix Operations for Partial Derivatives

#### Appendix A.5. Contents of the Partial Derivatives

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**Figure 3.**Different cases for the kinematic constraints of the leading chain for the 3RRR example: (

**a**) no constraints complied; (

**b**) position and tool axis rotation complied; (

**c**) all constraints complied.

**Figure 4.**Overview of the different frames (

**a**) for six-DoF tasks with standard Euler angle convention and (

**b**) for five-DoF tasks with reciprocal Euler angle convention; taken from [5].

**Figure 5.**Different cases for the kinematic constraints of the following chain: (

**a**) wrong position and orientation; (

**b**) correct position and wrong orientation; (

**c**) all kinematic constraints are complied.

**Figure 6.**

**Left**: Table with the kinematic parameters of the industrial manipulator Fanuc M-710 iC/50.

**Right**: sketch of the robot scenario.

**Figure 7.**Results of the inverse kinematics with different settings for the trajectory of Figure 6.

**Figure 8.**Results of the inverse kinematics of a 6UPS robot in a 3T2R task. (

**a**) platform positions, (

**b**) platform orientation in Euler angles, (

**c**,

**d**) optimization criteria ${h}_{1}\left(\mathit{q}\right)$ and ${h}_{2}\left(\mathit{q}\right)$.

**Figure 9.**Histograms with cumulated frequency of the IK success for all kinematic chains with different settings: 3T2R tasks (

**a**,

**b**) vs. 3T3R tasks (

**c**,

**d**) and arbitrary initial value (

**a**,

**c**) vs. initial near goal pose.

**Figure 10.**Detailed Statistics of the success of the inverse kinematics algorithm for 3T2R tasks (

**a**,

**b**) and 3T3R tasks (

**c**,

**d**). The Success of the IK solver is shown in different shades of green for increasing numbers of required tries. Different initial values ${\mathit{q}}^{0}$ are distinguished in (

**a**,

**c**) and (

**b**,

**d**).

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**MDPI and ACS Style**

Schappler, M.; Tappe, S.; Ortmaier, T. Modeling Parallel Robot Kinematics for 3T2R and 3T3R Tasks Using Reciprocal Sets of Euler Angles. *Robotics* **2019**, *8*, 68.
https://doi.org/10.3390/robotics8030068

**AMA Style**

Schappler M, Tappe S, Ortmaier T. Modeling Parallel Robot Kinematics for 3T2R and 3T3R Tasks Using Reciprocal Sets of Euler Angles. *Robotics*. 2019; 8(3):68.
https://doi.org/10.3390/robotics8030068

**Chicago/Turabian Style**

Schappler, Moritz, Svenja Tappe, and Tobias Ortmaier. 2019. "Modeling Parallel Robot Kinematics for 3T2R and 3T3R Tasks Using Reciprocal Sets of Euler Angles" *Robotics* 8, no. 3: 68.
https://doi.org/10.3390/robotics8030068