# Topological Analysis of a Novel Compact Omnidirectional Three-Legged Robot with Parallel Hip Structures Regarding Locomotion Capability and Load Distribution

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. The SLIP Model

#### 1.2. SLIP-Based Real World Locomotion

## 2. Methods

#### 2.1. Design and General Topology

#### 2.2. Internal Mechanical Design

#### 2.3. SPM-Based Walking

#### 2.4. The SPM as a Hip Joint

#### 2.5. Load Support Joint

#### 2.6. Mechanical Redundancy and Topology

#### 2.7. Materials and Prototype

#### Manufacturing and Mounting

#### 2.8. Legged Conceptual Prototype Model

#### 2.9. Simulation Framework and Setup

#### 2.9.1. Plant Model

#### 2.9.2. Controller Model

#### 2.9.3. Simulation Setup

`ode2`(Heun) at a fixed time step of $\Delta t=2.5\times {10}^{-5}$ s for the mechanical system was chosen, which was selected as the fastest configuration possible with robust and stable behaviour for the robot model. Larger step sizes for the numerical integration showed unstable behaviour occurring at the moment of collision impacts, caused by the sudden penetration of the foots into the ground plane, resulting in large peaks of ground reaction forces. Hence, for this particular model, a small integration step size is required without the alternative of using softer contacts with reduced stiffness and damping parameters, which helps to lower impact forces at larger step sizes but also restricts the simulation accuracy. Thus, the parameters were chosen to keep the deviation between the behaviour of the robot model inside the simulation and its assumed behaviour in the real-world small. Table 6 sums up important simulation parameters.

#### 2.10. Motion, Load and Actuation Analysis

#### 2.10.1. Motion Profiles

#### 2.10.2. Joint Load Measurement

#### 2.10.3. Controller Setup

#### 2.11. Walking Analysis

`torsoGoalReal`and feet

`footGoalFluent`. These values are updated permanently and depend on the walking direction and speed. Since walking is characterized as an alternating sequence of swing and stance states for each leg, based on a set of conditions

`A`, a value

`footGoalStatic`is derived form the

`footGoalFluent`value, working as a fixed target for one foot, being constant for a short period of time. The idea is to always pick a temporarily static value from the continuous target calculation, which is important in regard of providing a fixed target for the trajectory planning, thus yielding more stable motion profiles. Depending on a set of conditions

`B`, motion is then carried out for each leg in a forced (

`forceMotion = 1`) or non-forced (

`forceMotion = 0`) way. By dividing into these two motion types, the current state of each leg gets evaluated, and a priority is calculated, yielding the decision regarding which leg has to change its state to the swing mode, as a measure to keep the robot in a stable motion. In the case that no immediate need for the transition to the swing mode exists, the controller either tries to keep the leg in its last initial contact position

`contactPoint`at the collision impact with the ground plane, if in contact, or naturally moves to the next fixed

`footGoalStatic`position. For the situation that a leg is in the swing mode, a set of conditions

`C`is checked, which is used to cancel any swing motion or forced motion, before letting the robot become unstable. As a preliminary step regarding the current intended and actually reachable target position, the foot target is set to the value

`footGoalReal`inside the function-blocks handling the leg motion, before being used in the last step for the actual update of the desired robot trajectory. This final trajectory data are then forwarded into the next controller block for the calculation of the robot inverse kinematics solution.

`A`,

`B`and

`C`were evaluated by the state of the robot model in simulation. The parameters used for the decision process and the set of rules mainly consist of timing, force, position, speed, and distance information gathered in the previous sensor data interpreter step, which are updated at each simulation iteration and are compared against a manually tuned set of parameters.

## 3. Results

#### 3.1. Load Analysis Results

#### 3.2. Actuator Analysis Results

#### 3.3. Individual Linkage Observation

#### 3.4. General Comparison

#### 3.5. Walking Analysis Results

#### 3.6. Linkage Load Calculation

#### 3.6.1. Mathematical Derivation

#### 3.6.2. Special Case

#### 3.6.3. Visual Depiction of Isotropy

#### 3.6.4. Visual Depiction of Singularity

#### 3.6.5. Numerical Considerations

`inv()`operator, which calculates the matrix inverse by LU-decomposition, and operates on the matrices $\mathit{A}$ and $\mathit{B}$ from Equation (28); B also uses $\mathit{A}$ and $\mathit{B}$ from Equation (28) but actually solves the linear system of equation by the Matlab∖operator. C rearranges Equation (28) into the shortest possible single matrix expression, while D uses Equation (40). The Matlab code is available via the Supplementary Materials. As it becomes obvious, the specifically tailored expressions of C and D show the fastest calculation, which suggests that in an actual controller implementation, one must be careful regarding the Jacobian construction.

#### 3.6.6. Internal Linkage Stress

## 4. Discussion

## 5. Conclusions and Future Work

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**SLIP model for dynamically balanced running with body point mass m, spring stiffness k, and uncompressed leg length ${l}_{0}$.

**Figure 2.**General conceptual robot design overview. The feet are considered as points, colliding with an even ground plane.

**Figure 3.**Actuator placement inside the leg. Dashed lines mark influence of the individual actuators on the actual topological joints of the robot model.

**Figure 4.**General layout of a spherical parallel manipulator. Dotted lines indicate actuated (black) and passive (gray) rotatory axes ${\widehat{\mathit{u}}}_{j}$, ${\widehat{\mathit{v}}}_{j}$, ${\widehat{\mathit{w}}}_{j}$.

**Figure 5.**Typical 3RCC (

**a**) and altered (

**b**) topological non-redundant 3CCC+S layout of the SPM. The topology of the single linkage structure with proximal link $pl$ and distal link $dl$ can be observed for both layouts. Red marks actuated joints. In the case (

**b**), an additional joint with no DOF connects the tool-platform and an additional tcp body, which was included for load measurement purposes. Note that the tcp body in this model is the actual interface to external forces/torques or tools.

**Figure 6.**Real-world prototype (

**a**) of the hip joint mechanism based on stereolithography 3d printing and thin supporting aluminium structures. Encoder integration (

**b)**and visible screw holes for the torso connector structure on the bottom side of the manipulator unit.

**Figure 7.**Spherical ball and socket artificial hip joint (

**a**) made of ceramic by CeramTec [29]. Image (

**b**) shows mechanical details of the motor connection.

**Figure 8.**Interlocking radial symmetrical arrangement of the hip units around the vertical centre axis. Yellow circles mark the centre of the SPM units, and origins of the SPM centre reference frames. Red, green, and blue colours indicate x-, y-, and z-coordinates and individual hip units, respectively. White numbers relate to the individual branches, while black numbers identify the unit itself. The radius ${r}_{h}$ to the spherical centre of each hip joint is perpendicular to the vertical centre axis, which is the z-axis of the torso reference frame. SPM (3) is placed inside the y-z-plane of the torso reference frame.

**Figure 9.**Robot assembly with actual CAD models, showing the interlocking arrangement of the individual hip units.

**Figure 10.**CAD model of the robot in the simulation framework. Grid distances and colour mapping: 1 m (yellow), 1 dm (cyan), 1 cm (grey).

**Figure 14.**Full robot topology ${T}_{\u25b5}$ without (

**a**) and topology ${T}_{\u25b4}$ with supporting load joint (

**b**). Red-coloured connections are actuated (active) joints. Other joints are passive, non-actuated joints. Abbreviations are pl for proximal link, dl for distal link, ul for upper leg, and ll for lower leg.

**Figure 15.**Motion profiles and screenshots of the robot model in simulation, following the reference trajectory. The term with support relates to the configuration ${T}_{\u25b4}$ with load support joint and no support to the conf. ${T}_{\u25b5}$ without bespoke joint. The screenshots display extrema of the corresponding motion plots.

**Figure 16.**CAD model of the hip mechanism inside the specially purposed simulation framework for this project.

**Figure 17.**Reference frame view of the hip unit, showing base (red), link (orange), tool (yellow), hip (cyan), and tcp (magenta) joints. Cylinders visualize the joint axis direction. Magenta lines show connections between joints. Red, green, and blue axes represent the orthogonal x-, y-, and z-axes of the respective reference frames, respectively.

**Figure 19.**Load forces and torques in horizontal and vertical directions, measured at the hip and the tcp joint for both configurations ${T}_{\u25b4}$ and ${T}_{\u25b5}$.

**Figure 20.**Required actuator torques (

**a**) and velocities (

**b**) for both robot configurations and each motion profile A, B and C.

**Figure 21.**Simulated reaction forces (

**a**) and torques (

**b**) of each joint per linkage chain over time for motion profile B.

**Figure 22.**Angular joint distribution for different motion profiles, depicting the actuator joint angles and the resulting hip orientation in Euler angles. For the purpose of compact visualization, data for ${\varphi}_{1}$, ${\varphi}_{2}$, and ${\varphi}_{3}$ are shifted by $\Delta \varphi =-\frac{\pi}{2}$.

**Figure 23.**Reaction force and torque of the hip and the tcp joint in the horizontal and vertical component view. ${f}_{grf}$ shows ground reaction forces, and ${\tau}_{R,ui}$ displays the joint reaction torques after regarding actuator torques and joint friction. All values are absolute resp. norms of vector values. A different axis range is used for the walking profile.

**Figure 26.**Internal forces along the proximal and distal link for the EEF-torque ${\mathbf{\tau}}_{\mathit{eef}}=-1\phantom{\rule{4.pt}{0ex}}\mathrm{Nm}\phantom{\rule{0.166667em}{0ex}}{\widehat{\mathit{e}}}_{z}$. Link arc angles are set to ${\alpha}_{uv}={\alpha}_{vw}={60}^{\circ}$ for visualization purposes.

**Table 1.**Configuration parameters for different topologies. Abbreviations for joint DOFs are base (b), link (l), tool (t), hip (h), knee (k), foot–ground contact (f), and passive (p).

Config. | l | n | ${f}_{b}$ | ${f}_{l}$ | ${f}_{t}$ | ${f}_{h}$ | ${f}_{k}$ | ${f}_{f}$ | ${f}_{p}$ | F |

3SRS | 8 | 9 | - | - | - | 3 | 1 | 3 | 3 | 6 |

3RRR | 8 | 9 | 1 | 1 | 1 | - | - | - | 0 | −3 |

3RCC | 8 | 9 | 1 | 2 | 2 | - | - | - | 0 | 3 |

3CCC + S | 8 | 10 | 2 | 2 | 2 | 3 | - | - | 0 | 3 |

3CCC | 8 | 9 | 2 | 2 | 2 | - | - | - | 0 | 6 |

Parameter | ${\alpha}_{\mathit{uv}}$ | ${\alpha}_{\mathit{vw}}$ | ${\alpha}_{\mathit{us}}$ | ${\alpha}_{\mathit{sv}}$ | ${\alpha}_{\mathit{uu}}$ | ${\alpha}_{\mathit{ww}}$ |

Angle (deg) | 90 | 90 | $54.74$ | $54.74$ | 90 | 90 |

**Table 3.**Z-X-Z-Euler angles in degree of the individual hip centre frames in relation to the torso body reference frame.

Hip j | ${\mathit{Z}}_{0}$ | ${\mathit{X}}_{0}$ | ${\mathit{Z}}_{1}$ |
---|---|---|---|

1 | 120 | 150 | 5 |

2 | −120 | 150 | 5 |

3 | 0 | 150 | 5 |

**Table 4.**Properties of the physical robot model. N refers to the number of repeated assemblies in the full robot with part mass m.

Assembly | N | Mass (g) | $\mathit{N}{\displaystyle \frac{\mathit{m}}{\mathit{M}}}$ (%) |

Torso-connector-structure | 1 | 47.504 | 2.32 |

SPM base-platform | 3 | 354.199 | 51.84 |

SPM prox. link | 9 | 27.810 | 12.21 |

SPM dist. link | 9 | 10.009 | 4.39 |

SPM tool-platform | 3 | 28.701 | 4.20 |

Upper leg | 3 | 155.034 | 22.69 |

Lower leg | 3 | 15.984 | 2.34 |

Robot | 1 | 2049.629 | 100.00 |

Parameter | Value | Unit |
---|---|---|

Hip radius ${r}_{h}$ | $0.100$ | m |

Foot radius ${r}_{f}$ | $0.240$ | m |

Torso height ${h}_{t}$ | $0.510$ | m |

Upper leg length ${l}_{ul}$ | $0.200$ | m |

Lower leg length ${l}_{ll}$ | $0.300$ | m |

Parameter | Value | Unit |
---|---|---|

Floor stiffn. ${k}_{f}$ | 5000 | N/m |

Floor damp. ${d}_{f}$ | 20 | Ns/m |

Static fric. ${\mu}_{s}$ | 0.8 | - |

Dyn. fric. ${\mu}_{d}$ | 0.7 | - |

Velocity threshold $\epsilon $ | $1.0\times {10}^{-3}$ | m/s |

Plant step time $\Delta t$ | $2.5\times {10}^{-5}$ | s |

Contr. step time $\Delta t$ | $1.0\times {10}^{-3}$ | s |

Simulink solver | ode2 (Heun) | - |

Joint fric. coeff. ${\mu}_{j}$ | 0.01 | - |

Joint fric. damp. ${d}_{j}$ | ${10}^{-4}$ | Nm·s/rad |

Var. | Unit | Profile A | Profile B | Profile C |
---|---|---|---|---|

${h}_{t}$ | m | $0.444$ | $0.482$ | $0.485$ |

${r}_{f}$ | m | $0.246$ | $0.246$ | $0.246$ |

$\omega $ | $\frac{\mathrm{rad}}{\mathrm{s}}$ | $2\pi 0.250$ | $2\pi 0.125$ | $2\pi 0.250$ |

${r}_{\mathit{O}}$ | m | 0 | $0.040$ | 0 |

${x}_{\mathit{O}}$ | m | 0 | ${r}_{\mathit{O}}cos\left(\omega t\right)$ | 0 |

${y}_{\mathit{O}}$ | m | 0 | ${r}_{\mathit{O}}sin\left(\omega t\right)$ | 0 |

${z}_{\mathit{O}}$ | m | ${h}_{t}+0.05sin\left(\omega t\right)$ | ${h}_{t}$ | ${h}_{t}$ |

${\varphi}_{\mathit{O}}$ | deg | 0 | 0 | $15sin\left(\omega t\right)$ |

Variable | ${\mathit{r}}_{\mathit{base}}$ | ${\mathit{r}}_{\mathit{link}}$ | ${\mathit{r}}_{\mathit{tool}}$ | ${\mathit{r}}_{\mathit{tcp}}$ |
---|---|---|---|---|

Radius (m) | $0.052$ | $0.040$ | $0.035$ | $0.035$ |

Profile | ${\mathit{K}}_{\mathit{P},\mathit{b}}$ | ${\mathit{K}}_{\mathit{I},\mathit{b}}$ | ${\mathit{K}}_{\mathit{D},\mathit{b}}$ | ${\mathit{K}}_{\mathit{P},\mathit{k}}$ | ${\mathit{K}}_{\mathit{I},\mathit{k}}$ | ${\mathit{K}}_{\mathit{D},\mathit{k}}$ |
---|---|---|---|---|---|---|

A/B/C | 8.50 | 0.70 | 0.03 | 8.50 | 1.00 | 0.03 |

Walking | 11.00 | 0.00 | 0.07 | 14.00 | 0.00 | 0.06 |

**Table 10.**Load comparison for ${T}_{\u25b5}$ and ${T}_{\u25b4}$ with fixed actuator axes, depicting static tripodal standing.

Var. | ${\mathit{T}}_{\mathit{\u25b5}}$ | ${\mathit{T}}_{\mathit{\u25b4}}$ | Unit |
---|---|---|---|

$|{\mathit{f}}_{R}^{tcp}|$ | 5.025 | 0.769 | N |

$|{\mathbf{\tau}}_{R}^{tcp}|$ | 0.801 | 0.876 | Nm |

${\tau}_{u1}$ | −0.785 | Nm | |

${\tau}_{u2}$ | 0.371 | Nm | |

${\tau}_{u3}$ | 0.402 | Nm | |

${\tau}_{k}$ | 0.259 | Nm | |

${h}_{t}$ | 0.513 | m | |

${\varphi}_{1}$ | 125.7 | ${}^{\circ}$ | |

${\varphi}_{2}$ | 125.7 | ${}^{\circ}$ | |

${\varphi}_{3}$ | 125.7 | ${}^{\circ}$ | |

${\varphi}_{k}$ | −22.5 | ${}^{\circ}$ |

Load | A | B | C |
---|---|---|---|

${r}_{\mathit{f},\mathit{load}}$ | 0.84 | 0.84 | 0.86 |

${r}_{\tau ,\mathit{load}}$ | 0.05 | 0.17 | −0.43 |

**Table 12.**Construction parameters for ${\widehat{\mathit{u}}}_{j}$ and ${\widehat{\mathit{w}}}_{j}$ axes for the SPM home configuration and conditioning index $\eta $.

Parameter | Used | Ideal |
---|---|---|

${\gamma}_{uj}$ | $arccos\left(\sqrt{2/3}\right)\approx 35.{26}^{\circ}$ | |

${\gamma}_{wj}$ | ${\gamma}_{uj}$ | |

${\delta}_{uj}$ | ${0}^{\circ}$, ${120}^{\circ}$, ${240}^{\circ}$ | |

${\delta}_{wj}$ | ${\delta}_{uj}+{45}^{\circ}$ | ${\delta}_{uj}+{60}^{\circ}$ |

$\eta $ | 0.978 | 1.000 |

Leg | 1 | 2 | 3 |
---|---|---|---|

${r}_{s}$$(\%)$ | 96.7 | 96.5 | 96.7 |

**Table 14.**Average execution time on Intel i5-8365U CPU, based on $n={10}^{8}$ samples, measured for calculating the solution to Equation (39).

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**MDPI and ACS Style**

Feller, D.; Siemers, C. Topological Analysis of a Novel Compact Omnidirectional Three-Legged Robot with Parallel Hip Structures Regarding Locomotion Capability and Load Distribution. *Robotics* **2021**, *10*, 117.
https://doi.org/10.3390/robotics10040117

**AMA Style**

Feller D, Siemers C. Topological Analysis of a Novel Compact Omnidirectional Three-Legged Robot with Parallel Hip Structures Regarding Locomotion Capability and Load Distribution. *Robotics*. 2021; 10(4):117.
https://doi.org/10.3390/robotics10040117

**Chicago/Turabian Style**

Feller, David, and Christian Siemers. 2021. "Topological Analysis of a Novel Compact Omnidirectional Three-Legged Robot with Parallel Hip Structures Regarding Locomotion Capability and Load Distribution" *Robotics* 10, no. 4: 117.
https://doi.org/10.3390/robotics10040117