# Bio-Inspired Optimal Control Framework to Generate Walking Motions for the Humanoid Robot iCub Using Whole Body Models

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## Abstract

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## 1. Introduction

- Formulation of the whole-body dynamic model for walking problems for the humanoid robot iCub.
- Formulation of the optimal control problem for a complex walking sequence involving cyclic, as well as non-cyclic motion phases.
- The use of a direct multiple shooting method, which allows solving the equations of motion at a high precision at all times in the gait and to precisely take into account the kinematic and dynamic constraints of the robot.
- Implementation and experimental validation on the HeiCub humanoid, a reduced iCub, with computation of performance indicators as a comparison with motions generated using reduced models.

## 2. The (He)iCub Humanoid Robot

## 3. Model and Dynamics

#### 3.1. Walking Phases

- Starting step, the robot starts from a complete stop (i.e., all velocities to zero) and takes the first step, leading to the periodic motion.
- Periodic steps, which are the steps that the robot can repeat during walking. In this case, we assume single-step periodic, i.e., the left and right leg configurations can be mirrored, as the robot is symmetric. The periodicity is enforced on touchdown.
- Ending step, the final step where the robot comes to a complete stop from the periodic motion.

- DS: Double Support, where both feet are on the ground.
- LSS: Left Single Support, where the left foot is on the ground and the right leg swings to the next support location.
- RSS: Right Single Support, as for LSS, the right foot is on the ground and the left leg is swinging.

- RTD: Right Touch Down, when the left foot is in single support and the right foot strikes the ground, we assume that when the foot of the robot touches the ground, it is completely flat.
- LTD: Left Touch Down, the left foot strikes the ground when the right foot is in single support.

#### 3.2. Dynamics

## 4. Optimal Control Problem

#### 4.1. States, Controls and Parameters

#### 4.2. Constraints

- Joint angles range,
- Joint velocities,
- Torques.

#### 4.3. Objective Functions

- Minimization of joint torques squared, which is always included to ensure smooth torques (with small weighting factor):$${\mathsf{\Phi}}_{\mathit{\tau}}(\xb7)=\mathbf{W}\xb7{\mathbf{u}}^{T}\mathbf{u}.$$
- Minimization of absolute mechanical work:$${\mathsf{\Phi}}_{work}(\xb7)=\mathbf{W}\xb7\sum _{i=0}^{{n}_{dof}-6}\mid {\dot{q}}_{i-6}\xb7{u}_{i}\mid .$$
- Minimization of joint accelerations squared in order to obtain smooth velocity trajectories (with small weighting factor):$${\mathsf{\Phi}}_{\ddot{q}}(\xb7)=\mathbf{W}\xb7{\ddot{\mathbf{q}}}^{T}\ddot{\mathbf{q}}.$$
- Torso orientation minimization (ort. min., which is sometimes also referred to as torso stabilization) in terms of torso movements with respect to the world reference:$${\mathsf{\Phi}}_{torso}(\xb7)={\theta}_{torso}-{\theta}_{world}.$$

## 5. Results

#### 5.1. Software Tools

#### 5.2. Numerical Results

- Minimization of joint torques, with a small weight on minimization of joint accelerations.
- Minimization of joint torques and torso orientation minimization, with a small weight on minimization of joint accelerations.
- Minimization of absolute work, with a small weight on minimization of joint torques and joint accelerations.
- Minimization of absolute work and torso orientation, with a small weight on minimization of joint torques and joint accelerations.

## 6. Experimental Validation

#### Key Performance Indicators

- Walking velocity ${v}_{max}$:The maximum achieved walking velocity; in the case of motions generated with optimal control, this corresponds to the velocity of the resulting sequence.
- Walking timings ${t}^{ss}/{t}^{ds}$ and step period:Single- and double-support times of a single step, the whole duration of which is indicated as the step period. In the case of optimal control, we consider the timings of the periodic step.
- Cost of transport:$${\mathbf{E}}_{CT}=\frac{{\sum}_{m=1}^{M}{\int}_{{t}_{0}}^{{t}_{f}}{I}_{m}\left(t\right){V}_{m}\left(t\right)dt}{{m}_{robot}g\xb7d}$$The cost of transport is defined as a unitless quantity, where M is the total number of motors, ${I}_{m}$ and ${V}_{m}$ are the current and voltage measurements of the motor m, ${m}_{robot}$ is the mass of the robot and d is the traveled distance.
- Froude number:$$Fr={v}_{max}/\sqrt{g\xb7h},\phantom{\rule{4pt}{0ex}}\mathrm{for}\phantom{\rule{4pt}{0ex}}h={l}_{leg}$$
- Precision of task execution:Expressed in terms of tracking errors. The root mean square error is computed by summing the squared difference between the measurement and the desired position over all points of the whole trajectory.

## 7. Discussion and Conclusions

- It is possible to formulate optimal control problems for the iCub/HeiCub robots that allow one to simultaneously generate periodic walking motions, as well as the necessary starting and stopping steps that take the robot from standing position to the periodic cycle, as well as from back to standing position. So far, only single starting and stopping steps have been included in the formulation, but it is straightforward to extend this to multiple steps, which may be required for faster walking.
- Different objective functions result in visibly different walking styles for the robot. In particular we have compared a minimization of torques squared and of absolute mechanical work, in both cases with and without a combined term on torso stabilization. A very small term on joint accelerations was present in all objective functions. Further objective functions are the subject of current research. The fact that we are able to generate walking in different styles in an automated way already presents a significant difference from the classical walking generation methods using the simple models for which all outcomes look very similar.
- The resulting motions are still not close to biological motion, but they have made significant progress in the right direction. In particular, the motions show variations in the height of the CoM. See the discussion below for making optimized walking motions more biological or “human-like”.

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**The HeiCub (iCub of Heidelberg University) humanoid robot. In red, the series elastic actuators, which are not considered in the context of this work.

**Figure 2.**Walking phases of HeiCub. DS = Double Support, LSS = Left Single Support, RSS = Right Single Support, LTD = Left Touch Down, RTD = Right Touch Down. The whole sequence can be seen as three sub-sequences. The periodic step can be repeated for a desired number of times that do not need to be further modeled.

**Figure 3.**Feet shapes of the HeiCub robot. Three points are defined for contact modeling on each foot.

**Figure 5.**Approximation of the support area of the foot. The darker grey zones delimit the support areas for single support cases. The lighter grey zone changes according to the positions of the feet.

**Figure 6.**Example sequences of one step motions from using the reduced model, the whole-body model minimizing torques and the whole-body model with torso orientation minimization.

**Figure 7.**Center of mass trajectories of a full sequence with nine periodic steps combinations of objectives as in Table 2.

**Figure 8.**Joint angle trajectories of all the 15 DOF of the robot, obtained for the four combinations of objective functions. Time is normalized for all trajectories for comparison purposes. It is to be noted that torso orientation minimization does not mean reduction of the movements of the torso joints, but the orientation w.r.t. the world reference.

**Figure 9.**Torso orientation trajectories for the two cases of minimization of torques and absolute work. The orientation is expressed with respect to the world reference frame. The nominal reference (torso upright) is [90, 0, −90]. Time is normalized for all trajectories for comparison purposes.

**Figure 10.**[Hardware experiment result.] Center of mass trajectories obtained on the robot (in red) with all four combinations of objective functions as in Table 2.

Joint | Range Limits (deg) | Velocity Limits (deg/s) | Torque Limits (Nm/s) |
---|---|---|---|

l_hip_pitch, r_hip_pitch | [−33, 100] | [−100, 100] | [−50, 50] |

l_hip_roll, r_hip_roll | [−19, 90] | [−150, 150] | [−50, 50] |

l_hip_yaw, r_hip_yaw | [−75, 75] | [−150, 150] | [−50, 50] |

l_knee, r_knee | [−100, 0] | [−150, 150] | [−50, 50] |

l_ankle_pitch, r_ankle_pitch | [−36, 27] | [−150, 150] | [−50, 50] |

l_ankle_roll, r_ankle_roll | [−24, 24] | [−150, 150] | [−50, 50] |

torso_pitch | [−20, 60] | [−150, 150] | [−50, 50] |

torso_roll | [−26, 26] | [−150, 150] | [−50, 50] |

torso_yaw | [−50, 50] | [−150, 150] | [−50, 50] |

**Table 2.**Combination of objective functions. Minimization of torques and joint accelerations is always included with small weighting factors to ensure smooth trajectories.

Objectives | ${\mathbf{\Phi}}_{\mathit{\tau}}(\xb7)$ | ${\mathbf{\Phi}}_{\mathit{work}}(\xb7)$ | ${\mathbf{\Phi}}_{\ddot{\mathit{q}}}(\xb7)$ | ${\mathbf{\Phi}}_{\mathit{torso}}(\xb7)$ |
---|---|---|---|---|

1 | 1 | 0 | ${10}^{-4}$ | 0 |

2 | 1 | 0 | ${10}^{-4}$ | 10 |

3 | ${10}^{-4}$ | 1 | ${10}^{-4}$ | 0 |

4 | ${10}^{-4}$ | 1 | ${10}^{-4}$ | 10 |

**Table 3.**[Hardware experiment result.] Key Performance Indicators (KPIs) measured for all 4 combinations of objective functions, in comparison with results with reduced models. The support times and step period refer to the ones of the periodic step. NMPC, Nonlinear Model Predictive Control.

KPIs | Cart-Table | NMPC (LIPM) | Min Torques | Min Torques and Torso ort. | Min Work | Min Work and Torso ort. |
---|---|---|---|---|---|---|

${v}_{max}$ (m/s) | 0.037 | 0.065 | 0.053 | 0.079 | 0.043 | 0.053 |

${t}^{ss}/{t}^{ds}$ (s) | 1.5/1.0 | 0.6/0.6 | 1.06/0.98 | 0.7/0.37 | 1.06/1 | 0.8/0.8 |

step period (s) | 2.5 | 1.2 | 2.04 | 1.07 | 2.06 | 1.6 |

Cost of Transport | 4.27 | 2.99 | 2.69 | 1.97 | 3.32 | 3.09 |

Froude Number | 0.017 | 0.029 | 0.024 | 0.035 | 0.019 | 0.024 |

Joint error (deg) | 1.45 | 1.21 | 1.27 | 1.29 | 1.11 | 0.9 |

CoM error (cm) | 0.61 | 0.44 | 0.29 | 0.31 | 0.39 | 0.29 |

KPIs | Cart-Table | NMPC (LIPM) | Min Torques | Min Torques and Torso ort. | Min Work | Min Work and Torso ort. |
---|---|---|---|---|---|---|

Cost of Transport | 3.99 | 2.54 | 2.49 | 2.30 | 2.56 | 2.67 |

Joint error (deg) | 1.35 | 1.12 | 1.19 | 1.22 | 1.08 | 0.9 |

CoM error (cm) | 0.59 | 0.42 | 0.25 | 0.28 | 0.36 | 0.27 |

**Table 5.**[Hardware experiment result.] Key Performance Indicators (KPIs) for the periodic step. Note that in the case of motions obtained with the cart-table and NMPC, no explicit periodicity constraints were introduced, so the data are approximated on an intermediate step that would correspond to a periodic step.

KPIs | Cart-Table | NMPC (LIPM) | Min Torques | Min Torques and Torso ort. | Min Work | Min Work and Torso ort. |
---|---|---|---|---|---|---|

Cost of Transport | 4.33 | 3.10 | 2.89 | 1.90 | 3.31 | 3.19 |

Joint error (deg) | 1.65 | 1.20 | 1.37 | 1.39 | 1.20 | 1.0 |

CoM error (cm) | 0.63 | 0.46 | 0.31 | 0.35 | 0.40 | 0.32 |

KPIs | Cart-Table | NMPC (LIPM) | Min Torques | Min Torques & Torso ort. | Min Work | Min Work & Torso ort. |
---|---|---|---|---|---|---|

Cost of Transport | 4.05 | 2.64 | 2.54 | 2.20 | 2.61 | 2.62 |

Joint error (deg) | 1.35 | 1.15 | 1.20 | 1.20 | 1.08 | 0.85 |

CoM error (cm) | 0.61 | 0.43 | 0.26 | 0.30 | 0.38 | 0.27 |

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**MDPI and ACS Style**

Hu, Y.; Mombaur, K.
Bio-Inspired Optimal Control Framework to Generate Walking Motions for the Humanoid Robot iCub Using Whole Body Models. *Appl. Sci.* **2018**, *8*, 278.
https://doi.org/10.3390/app8020278

**AMA Style**

Hu Y, Mombaur K.
Bio-Inspired Optimal Control Framework to Generate Walking Motions for the Humanoid Robot iCub Using Whole Body Models. *Applied Sciences*. 2018; 8(2):278.
https://doi.org/10.3390/app8020278

**Chicago/Turabian Style**

Hu, Yue, and Katja Mombaur.
2018. "Bio-Inspired Optimal Control Framework to Generate Walking Motions for the Humanoid Robot iCub Using Whole Body Models" *Applied Sciences* 8, no. 2: 278.
https://doi.org/10.3390/app8020278