# Efficient Closed-Form Task Space Manipulability for a 7-DOF Serial Robot

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## Abstract

**:**

## 1. Introduction

#### 1.1. Contribution

- a new parametrization of the state- and null space that results in concise IK expressions with symmetric structure in the individual components
- analytical closed-form expressions from task space to manipulability measure w.r.t. joint limits, which allow array operation in vector-optimized programming languages. Note that array operation is also called Vectorization in e.g., MATLAB. It refers to the exploitation of Single Instruction Multiple Data (SIMD) instructions of modern Central Processing Unit (CPUs) and allows to operate on multiple data points simultaneously.
- sensitivity analysis of manipulability in task space
- real-time capable application for evaluating the task space manipulability of the entire null space, for globally optimal redundancy resolution w.r.t. manipulability of single poses and full trajectories on SE(3)

#### 1.2. Related Work

#### 1.2.1. Performance Measures

#### 1.2.2. Inverse Kinematics

#### 1.2.3. Optimizing Manipulability

#### 1.3. Outline

## 2. Problem Formulation

**Problem****1:**- Find a parametrization of the task- and null space that exploits the kinematic structure for concise expressions.
**Problem****2:**- Find closed-form expressions for all mappings from task space to manipulability that allow efficient array operation in vector-optimized programming languages.
**Problem****3:**- Let $\mathcal{Q}\subset {\mathbb{R}}^{7}$ be the space of admissible joint configurations. Find an analytical expression of the range of the null space solutions $\mathsf{\Lambda}\left(\mathit{z}\right):=\{\mathit{\lambda}\in {\mathbb{R}}^{n-6}\phantom{\rule{0.277778em}{0ex}}|\phantom{\rule{0.277778em}{0ex}}\mathrm{IK}(\mathit{z},\mathit{\lambda})\in \left(Q\right)\}$, for which the inverse kinematics function $\mathrm{IK}(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}},\mathit{\lambda})$ results in an admissible joint configuration $\mathit{q}\in \mathcal{Q}$.

## 3. Technical Approach

#### 3.1. Manipulability Measure

#### 3.1.1. Reduction of First Joint

**Proof.**

#### 3.1.2. Reduction of Last Joint

#### 3.1.3. Closed-Form Expression

#### 3.2. Task Space Parametrization

- PR1:
- uniquely define the null space parameter for the entire space of SE(3).
- PR2:
- result in a minimal number of parameters for the components of the IK vector map $\mathit{p}\mapsto \mathit{q}$.
- PR3:
- allow direct application of the above-mentioned reductions.

#### 3.2.1. Task Space Projection

#### 3.2.2. Task Space Surjection

#### 3.3. Inverse Kinematics

#### 3.3.1. Manipulator Configuration

#### 3.3.2. Elbow Angles

#### 3.3.3. Shoulder Angles

#### 3.3.4. Wrist Angles

#### 3.3.5. Overview

#### 3.4. Forward Kinematics

#### 3.5. Admissible Parameter Space

#### 3.5.1. Shoulder-Wrist Distance ${r}_{\mathrm{ref}}$

#### 3.5.2. Null Space Parameter $\lambda $

## 4. Results

#### 4.1. Accuracy

#### 4.2. Run-Time Comparison

^{7}samples, vectorization enables an even 50 times faster computation, compared to the implementation using for-loops. The advantage of calculating the task space manipulability on a GPU starts at an amount of 10

^{5}sample points. For a smaller number of samples, the overhead of initializing the data on the GPU does not pay off. Processing 10

^{7}samples, calculations on the GPU are 10 times faster then vectorized treatment on the CPU, and even 700 times faster than for conventional loop structures.Note that all time measurements include the generation of random samples on the CPU and GPU respectively.

#### 4.3. Sampling in Task Space

^{7}random samples. It shows that random sampling in joint space according to (61) is more likely to result in a joint configuration with poor manipulability of the robot. Uniform sampling in parameter space (63) produces much fewer joint configurations with poor manipulability, while at the same time more configurations with high manipulability. Naive sampling in parameter space (61) performs similarly good in the low manipulability section. However, it produces also fewer configurations with high manipulability. Considering a conventional 6-DOF robot, i.e., fixing the null space parameter $\lambda $ to 0 or $\pi $, results in a slightly better probability density function (PDF) than for the discussed 7-DOF mechanism. This is a surprising result, as it is always argued that the redundancy improves manipulability. While it is true that the additional DOF has the potential to improve performance measures, poor exploitation might achieve the opposite.

#### 4.4. Parameter Sensitivity Analysis of Manipulability in Parameter Space

^{7}random samples according to (63). These samples represent a uniform distribution of task space configurations. Figure 8 shows the bi-variate histograms of manipulability $\mu ({\mathit{p}}^{\mathrm{red}},\lambda )$ w.r.t. to the individual parameters.

#### 4.4.1. Translation Parameters ${r}_{\mathrm{ref}}$ and ${\beta}_{\mathrm{ref}}$

#### 4.4.2. Orientation Parameters ${\gamma}_{\mathrm{EE}}$ and ${\beta}_{\mathrm{EE}}$

#### 4.4.3. Null Space Parameter $\lambda $

#### 4.4.4. Discussion of Manipulability in Different Sampling Strategies

#### Naive vs. Uniformly Distributed Sampling

#### 6-DOF vs. 7-DOF Kinematics

#### 4.5. Number of Local Optima

## 5. Applications

#### 5.1. Optimal Robot Placement

#### 5.1.1. Best Overall Robot Configuration

#### 5.1.2. Best Robot Configuration for Multiple Task Poses

#### 5.1.3. Optimizing Robot Mounting Positions Regarding a Workspace Envelope

#### 5.2. Redundancy Resolution

#### 5.2.1. Redundancy Resolution for Global Manipulability Optima

#### 5.2.2. Optimizing Null Space Solution of Given End-Effector Trajectory

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CDF | cumulative distribution function |

CLIK | Closed-Loop Inverse Kinematic |

CPU | Central Processing Unit |

DOF | degree of freedom |

FK | Forward Kinematic |

GPU | Graphics Processing Unit |

IK | Inverse Kinematic |

M | Manipulability |

probability density function | |

PR | parameter requirements |

SIMD | Single Instruction Multiple Data |

S-R-S | Spherical-Revolute-Spherical |

TCP | tool center point |

TSP | Task Space Projection |

TSS | Task Space Surjection |

## Appendix A. Manipulability

## Appendix B. Inverse Kinematic Functions from (43)

## Appendix C. Absolute Valued Inverse Kinematics Functions from (55a)

## Appendix D. Admissible Null Space Parameter Functions from (56)

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**Figure 1.**Illustration of the task space manipulability at a given end-effector pose. The null space of this 7-DOF S-R-S kinematics consists of the free elbow position (joint 4) along a circle. This position defines the direction of the forearm, i.e., the vector from the shoulder to the wrist. The colored fan shows all possible forearm poses with the corresponding manipulability color-coded from dark red (very bad) to light green (optimal). Colorless areas of the fan mark areas that violate joint constraints.

**Figure 2.**Relation of task space $\mathit{z}$, parameter space $\mathit{p}$, joint space $\mathit{q}$, and manipulability metric $\mu $. The mappings are referred to as Task Space Projection (TSP) and Task Space Surjection (TSS), Forward Kinematic (FK) and Inverse Kinematic (IK), and Manipulability (M).

**Figure 3.**Parametrization of the Task Space. Positions of Base B and Shoulder S are fixed. Translation reference parameters $({r}_{\mathrm{ref}},{\gamma}_{\mathrm{ref}},{\beta}_{\mathrm{ref}})$ define the position of the Wrist W. The end-effector parameters $({\gamma}_{\mathrm{EE}},{\beta}_{\mathrm{EE}},{\psi}_{\mathrm{EE}})$ describe the rotation from reference frame R to tool frame T as consecutive $Z\to {Y}^{{}^{\prime}}\to {Z}^{{}^{\u2033}}$ Euler angles. The null space is parametrized with $\lambda $. It defines the position of the elbow E via relative rotation between the elbow oriented frame L and frame R.

**Figure 4.**Reference frames and their relations. The blue frames B to T are fixed to the corresponding body-fixed coordinate systems of the robot links. Orange frames R and L are additional reference frames for the introduced parameter space. The arrows mark the rotations between the frames of reference.

**Figure 5.**Run-time comparison of processing N poses w.r.t. their task space manipulability. Considered are the MATLAB robotics IK solver based on nonlinear optimization, the analytical IK solver by Shimizu et al. [15], and the presented approach in three versions: a conventional sequential loop structure, as well as vectorized evaluation on the central processing unit (CPU) and graphics processing unit (GPU).

**Figure 6.**Uniform distributed sampling of the task space (2000 samples). (

**a**) End-effector translation; (

**b**) End-effector orientation.

**Figure 7.**Approximated cumulative distribution function (CDF) from a histogram of manipulability w.r.t. different sampling strategies (10

^{7}samples).

**Figure 8.**Bi-variate histograms of $\mu ({r}_{\mathrm{ref}},{\beta}_{\mathrm{ref}},{\gamma}_{\mathrm{EE}},{\beta}_{\mathrm{EE}},\lambda )$ w.r.t. to the individual parameters, based on 10

^{7}uniformly distributed parameter space samples. Colors are normalized along with the particular value of the parameter on the x-axis.

**Figure 9.**Results of the task space manipulability optimization of a robot mounting pose. (

**a**): Overall best robot configuration. There are a total of 8 global optima with equal manipulability ${\mu}_{\mathrm{max}}=0.143$. From 1000 random initial starting points, 83% of the optimization runs converged to one of the global optima. (

**b**): Optimizing relative pose w.r.t. a workspace envelope of size $(\Delta x,\Delta y,\Delta z)=(0.4,0.4,0.3)\mathrm{m}$. Note that the cubic volume is projected onto the parameter space, hence the distortion in the illustration. The resulting configuration for pose ${\mathit{z}}_{0}$, again lies fully on the $xz$-plane. However, unlike the single best pose, only one single optimum is found.

**Figure 10.**Multiple local optima of manipulability $\mu $ in the null space of ${\mathit{p}}^{\mathrm{red}}={[0.6,0.7,1.4,0.7]}^{\top}$.

**Figure 11.**Null space manipulability over a parameter trajectory. The 3D plot

**(a)**shows an exemplary start $\mathit{p}(s=0)$ and end configuration $\mathit{p}(s=1)$. The contour plot

**(b)**shows the manipulability $\mu (\mathit{p},\lambda )$ of the full null space. Red lines mark the limits $\lambda (\mathit{p},{q}_{\mathrm{max}})$ of the admissible null space region. The numbers refer to the invoking joint. Blue circles mark desired ${\lambda}^{d}(s=0)$ and ${\lambda}^{d}(s=1)$, and the blue line marks a trajectory as it would be chosen by local optimization of ${\lambda}_{\mathrm{max}}\left(\mathit{p}\right)$.

# optima | 1 | 2 | 3 | 4 |
---|---|---|---|---|

percentage | 20% | 41% | 27% | 12% |

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**MDPI and ACS Style**

Huber, G.; Wollherr, D.
Efficient Closed-Form Task Space Manipulability for a 7-DOF Serial Robot. *Robotics* **2019**, *8*, 98.
https://doi.org/10.3390/robotics8040098

**AMA Style**

Huber G, Wollherr D.
Efficient Closed-Form Task Space Manipulability for a 7-DOF Serial Robot. *Robotics*. 2019; 8(4):98.
https://doi.org/10.3390/robotics8040098

**Chicago/Turabian Style**

Huber, Gerold, and Dirk Wollherr.
2019. "Efficient Closed-Form Task Space Manipulability for a 7-DOF Serial Robot" *Robotics* 8, no. 4: 98.
https://doi.org/10.3390/robotics8040098