# Integrated Structure-Control Design of a Bipedal Robot Based on Passive Dynamic Walking

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. System Description

#### 2.1. SPBR Dynamic Model

#### 2.1.1. Single Support Phase

#### 2.1.2. Double Support Phase

#### 2.1.3. Semi-Passive Control Strategy

## 3. Integrated Structure-Control Design Problem

#### 3.1. Objective Function

#### 3.1.1. Walking Capability Design Objective

#### 3.1.2. Energetic Efficiency Design Objective

#### 3.1.3. General S-C Design Objective

#### 3.2. Design Variables

#### 3.2.1. Structural Design Variables

#### 3.2.2. Semi-Passive Control Design Variables

#### 3.2.3. Walking Conditions Variables

#### 3.3. Constraints

#### 3.4. Optimization Problem Formulation

## 4. Results and Discussion

#### 4.1. Integrated S-C Design

#### 4.2. Integrated Structure-Control Design versus Sequential Design

- Integrated S-C design obtains a better synergy between walking capability and energetic efficiency design objectives than the sequential design method. This is demonstrated by assuming that a better synergy between criteria is when the trade-off in the aggregate function $J\left({p}^{\ast}\right)$ is minimum, meaning that waking capability and energetic efficiency are in equilibrium, the proposal reduces around $63.55\%$ the value of the aggregate function $J\left({p}^{\ast}\right)$ with respect to the sequential design (based on the column $J\left({p}^{\ast}\right)$ of Table 8). The overall behavior of both designs is shown in Figure 8. Both designs can develop a gait cycle in the two walking stages, i.e., the walking capability ${J}_{1}\left({p}^{\ast}\right)$ is suitable in both designs. Nevertheless, the proposal reduces $95.41\%$ of the control activation and its magnitude with respect to the sequential design based on the values reported in the column ${J}_{2}\left({p}^{\ast}\right)$ of Table 8.
- In both design approaches, the proposed semi-passive control strategy can produce a periodic movement of the SPBR into the SPWS. This achieves the same dynamic coupling between the frontal plane movement and the gait periods in both the PWS and SPWS of the sagittal plane (the dynamic coupling is reached in the oscillation periods ${T}_{fro}={T}_{sag}^{p}={T}_{sag}^{sp}=1.2s$, as indicates in Table 7). Thus, the walking capability for integrated and sequential design is assured.
- The semi-passive control signal is activated around 5% of the SPWS time in the integrated design (see Table 6 and Figure 8c). Meanwhile, in the sequential design, the semi-passive control strategy is activated around $89\%$ of the SPWS time (see Figure 8d). The features of the achieved semi-passive control signal in the integrated design are attributed to the high amplitude of the potential and kinetic energies oscillation (see Figure 10). This contributes to keeping the dynamic response of the SPBR into the corresponding limit cycle for a long time without the control influence. Hence, this results in the reduction of the control system activation.
- Based on the structure of the SPBR, the center of mass of integrated and sequential design is located almost at the same height with respect to the leveled ground (see columns $C{M}_{fro}^{y}$ and $C{M}_{sag}^{y}$ of Table 7). Nevertheless, the value of mass related to the integrated design is higher in comparison with the sequential one. In addition, it is observed that integrated S-C design approach exploit the structural properties of the SPBR to promote higher angular displacements along the PWS and SPWS. This fact is shown in Table 9, where the ratio between the maximum angular displacements and the permitted rolling angles in frontal and sagittal plane of the SPBR are $max\left(\right)open="("\; close=")">\left(\right)open="|"\; close="|">{\theta}_{f}\left(t\right)\approx 0.81{\beta}_{fro}$ and $max\left(\right)open="("\; close=")">\left(\right)open="|"\; close="|">{\theta}_{st}\left(t\right)$, respectively. Meanwhile, for the sequential design these relationships are $max\left(\right)open="("\; close=")">\left(\right)open="|"\; close="|">{\theta}_{f}\left(t\right)\approx 0.33{\beta}_{fro}$ and $max\left(\right)open="("\; close=")">\left(\right)open="|"\; close="|">{\theta}_{st}\left(t\right)$, respectively. Furthermore, the amplitude of the movement in frontal plane of the S-C design is associated with the value of ${R}_{f}^{\ast}$ and ${\theta}_{fi}^{\ast}$, where in both cases, integrated design approach provides higher values for these design variables. With respect to the sagittal plane, although the best design of each approach converged towards the same value of inclination angle ${\gamma}^{\ast}$ in the PWS, the structure and control parameters of the integrated one induced that the angular displacement and velocity are higher in both walking scenarios in comparison with the sequential design (see Table 9 and Figure 7). Therefore, the natural dynamics of the SPBR structure promotes a suitable recovery between the kinetic and potential energy to maintain the limit cycle with minimum control effort.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Nomenclature

Symbol | Definition |
---|---|

General | |

SPBR | Semi-Passive Bipedal Robot |

S-C | Structure-control design |

PWS | Passive Walking Stage |

SPWS | Semi-Passive Walking Stage |

${M}_{\sigma}$ | Inertia matrix |

${C}_{\sigma}$ | Centrifugal and Coriolis matrix |

${G}_{\sigma}$ | Gravity forces vector |

$\eta \in \left(\right)open="\{"\; close="\}">f,s$ | Generalized coordinates assignment |

$\sigma \in \left(\right)open="\{"\; close="\}">ss,rm,fc$ | SPBR behavior assignment |

$\varsigma \in \left(\right)open="\{"\; close="\}">f,s$ | SPBR plane assignment |

g | Acceleration due to gravity |

Sagittal plane [14] | |

${m}_{sag}$ | Mass of SPBR leg |

${I}_{sag}$ | Inertia moment of SPBR leg |

${R}_{s}$ | Foot radius |

b | Distance between leg mass center and the center of ${R}_{s}$ |

d | Distance between hip and the center of ${R}_{s}$ |

$\gamma $ | Slope angle |

${\theta}_{st},{\dot{\theta}}_{st}$ | Angular position and velocity of stance leg |

${\theta}_{sw},{\dot{\theta}}_{sw}$ | Angular position and velocity of swing leg |

${T}_{sag}^{p}$, ${T}_{sag}^{sp}$ | Sagittal plane gait period (PWS and SPWS) |

${\Omega}^{-}$, ${\Omega}^{+}$ | Pre-impact and post-impact angular momentum matrices |

$ss$ | Single support phase of sagittal plane |

${\delta}_{sag}$ | Foot rolling angle |

Frontal plane [14] | |

${m}_{fro}$ | Total mass of the SPBR |

${I}_{fro}$ | Inertia moment of the SPBR |

${R}_{f}$ | Foot radius |

a | Distance between mass center and the center of ${R}_{f}$ |

$\varphi $ | Half of feet separation angle |

${\theta}_{f},{\dot{\theta}}_{f}$ | Angular position and velocity of the SPBR |

${T}_{fro}$ | Oscillation period |

${\dot{{\theta}_{f}}}^{-},{\dot{{\theta}_{f}}}^{+}$ | Angular velocity before and after collision event |

$rm$ | Foot rolling movement |

$fc$ | Fixed kinematic chain behavior |

${\beta}_{fro}$ | Foot rolling angle |

Symbol | Definition |
---|---|

Parametrization | |

p | Design vector |

${p}_{s}$ | Vector of structure design variables |

${p}_{sg}$ | Vector of geometric design variables |

${p}_{sm}$ | Vector of material assignment variables |

${p}_{u}$ | Vector of semi-passive control design variables |

${p}_{\tau}$ | Vector of torque magnitudes |

${p}_{\Delta t}$ | Vector of activated control time intervals |

${p}_{wc}$ | Vector of walking condition variables |

${\tau}_{{k}_{s}}$ | Torque magnitude |

$\Delta {t}_{{k}_{s}}^{u}$ | Activated control time interval |

$\left(\right)$ | Foot-F geometric parameters |

$\left(\right)$ | Ankle-A geometric parameters |

$\left(\right)$ | Leg-L geometric parameters |

$\left(\right)$ | Hip-H geometric parameters |

$\left(\right)$ | Motor Case-MC geometric parameters |

$\left(\right)$ | Bearing-B geometric parameters |

$\left(\right)$ | Coupler-C geometric parameters |

$\left(\right)$ | Foot-F, Ankle-A, Leg-L, Hip-H and Motor Case-MC densities |

${\theta}_{fi}$ | Initial condition of frontal plane simulation |

Optimization problem | |

J | Aggregate function |

${J}_{1}$ | Walking capability design objective |

${J}_{2}$ | Energetic efficiency design objective |

${\Psi}_{f}$ | Difference of Poincaré mapping values (${\dot{\theta}}_{f}$) |

${\Psi}_{{s}_{1}}$ | Difference of Poincaré mapping values (${\theta}_{q}$) |

${\Psi}_{{s}_{2}}$ | Difference of Poincaré mapping values (${\dot{\theta}}_{q}$) |

${\Psi}_{{u}_{1}}$ | Measure of torque along SPWS |

${\Psi}_{{u}_{2}}$ | Measure of activated control time along SPWS |

$\left(\right)$ | Aggregate function weights |

${Q}_{\varsigma}$ | Poincaré Map |

${t}_{c}$ | Collision instant time |

${k}_{f}$ | Collision counter of frontal plane |

${k}_{s}$ | Collision counter of sagittal plane |

${t}_{fro}$, ${t}_{sag}$ | Frontal and sagittal dynamics simulation time |

Differential algorithm | |

${G}_{max}$ | Maximum number of generations (stop criterion) |

$NP$ | Population size |

$CR$ | Crossover factor |

K,F | Scale factors |

## References

- Westervelt, E.R.; Grizzle, J.W.; Chevallereau, C.; Choi, J.H.; Morris, B. Feedback Control of Dynamic Bipedal Robot Locomotion; CRC Press: Boca Raton, FL, USA, 2007; Volume 28. [Google Scholar]
- Wisse, M. Essentials of Dynamic Walking; Analysis and Design of Two-Legged Robots. Ph.D. Thesis, Delft University of Technology, Delft, The Netherlands, September 2004. [Google Scholar]
- Sakagami, Y.; Watanabe, R.; Aoyama, C.; Matsunaga, S.; Higaki, N.; Fujimura, K. The intelligent ASIMO: System overview and integration. In Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, Lausanne, Switzerland, 30 September–4 October 2002; Volume 3, pp. 2478–2483. [Google Scholar]
- Kaneko, K.; Harada, K.; Kanehiro, F.; Miyamori, G.; Akachi, K. Humanoid robot HRP-3. In Proceedings of the 2008 IEEE/RSJ International Conference on Intelligent Robots and Systems, Nice, France, 22–26 September 2008; pp. 2471–2478. [Google Scholar]
- Stasse, O.; Verrelst, B.; Vanderborght, B.; Yokoi, K. Strategies for humanoid robots to dynamically walk over large obstacles. IEEE Trans. Robot.
**2009**, 25, 960–967. [Google Scholar] [CrossRef] - Collins, S.; Ruina, A.; Tedrake, R.; Wisse, M. Efficient bipedal robots based on passive-dynamic walkers. Science
**2005**, 307, 1082–1085. [Google Scholar] [CrossRef] [PubMed][Green Version] - McGeer, T. Passive dynamic walking. J. Robot. Res.
**1990**, 9, 62–82. [Google Scholar] [CrossRef] - Wisse, M.; Keliksdal, G.; Van Frankenhyyzen, J.; Moyer, B. Passive-based walking robot. IEEE Robot. Autom. Mag.
**2007**, 14, 52–62. [Google Scholar] [CrossRef] - Collins, S.H.; Ruina, A. A bipedal walking robot with efficient and human-like gait. In Proceedings of the 2005 IEEE International Conference on Robotics and Automation, Barcelona, Spain, 18–22 April 2005; pp. 1983–1988. [Google Scholar]
- Ebrahimi, A.; Heydari, M.; Alasty, A. Active control of a passive bipedal walking robot. Int. J. Dyn. Control
**2017**, 5, 733–740. [Google Scholar] [CrossRef] - Kajita, S.; Yamaura, T.; Kobayashi, A. Dynamic walking control of a biped robot along a potential energy conserving orbit. IEEE Trans. Robot. Autom.
**1992**, 8, 431–438. [Google Scholar] [CrossRef] - Tedrake, R.; Zhang, T.W.; Fong, M.F.; Seung, H.S. Actuating a simple 3D passive dynamic walker. In Proceedings of the IEEE International Conference on Robotics and Automation, New Orleans, LA, USA, 26 April–1 May 2004; Volume 5, pp. 4656–4661. [Google Scholar]
- Bhounsule, P.A.; Cortell, J.; Grewal, A.; Hendriksen, B.; Karssen, J.D.; Paul, C.; Ruina, A. Low-bandwidth reflex-based control for lower power walking: 65 km on a single battery charge. Int. J. Robot. Res.
**2014**, 33, 1305–1321. [Google Scholar] [CrossRef] - Martínez-Castelán, J.N.; Villarreal-Cervantes, M.G. Frontal-Sagittal Dynamic Coupling in the Optimal Design of a Passive Bipedal Walker. IEEE Access
**2018**, 1. [Google Scholar] [CrossRef] - Pantoja-García, J.S.; Villarreal-Cervantes, M.G.; García-Mendoza, C.V.; Silva-García, V.M. Synergistic Design of the Bipedal Lower-Limb through Multiobjective Differential Evolution Algorithm. Math. Probl. Eng.
**2019**, 2019, 2301714. [Google Scholar] [CrossRef] - Villarreal-Cervantes, M.G.; Pantoja-García, J.S.; Rodríguez-Molina, A.; Benitez-Garcia, S.E. Pareto optimal synthesis of eight-bar mechanism using meta-heuristic multi-objective search approaches: Application to bipedal gait generation. Int. J. Syst. Sci.
**2021**, 52, 671–693. [Google Scholar] [CrossRef] - Tsuge, B.Y.; Plecnik, M.M.; McCarthy, J.M. Homotopy directed optimization to design a six-bar linkage for a lower limb with a natural ankle trajectory. J. Mech. Robot.
**2016**, 8, 061009. [Google Scholar] [CrossRef][Green Version] - Huang, Y.; Wang, Q.; Xie, G.; Wang, L. Optimal mass distribution for a passive dynamic biped with upper body considering speed, efficiency and stability. In Proceedings of the Humanoids 2008—8th IEEE-RAS International Conference on Humanoid Robots, Daejeon, Korea, 1–3 December 2008; pp. 515–520. [Google Scholar]
- Hale, A.L.; Dahl, W.; Lisowski, J. Optimal simultaneous structural and control design of maneuvering flexible spacecraft. J. Guid. Control. Dyn.
**1985**, 8, 86–93. [Google Scholar] [CrossRef] - Asada, H.; Park, J.H.; Rai, S. A control-configured flexible arm: Integrated structure control design. In Proceedings of the Robotics and Automation, Sacramento, CA, USA, 9–11 April 1991; pp. 2356–2362. [Google Scholar]
- Zhang, Y.; Yang, D.; Li, S. An integrated control and structural design approach for mesh reflector deployable space antennas. Mechatronics
**2016**, 35, 71–81. [Google Scholar] [CrossRef] - Bastos, G. A Synergistic Optimal Design for Trajectory Tracking of Underactuated Manipulators. J. Dyn. Syst. Meas. Control
**2019**, 141, 021015. [Google Scholar] [CrossRef] - Sands, T. Optimization Provenance of Whiplash Compensation for Flexible Space Robotics. Aerospace
**2019**, 6, 93. [Google Scholar] [CrossRef][Green Version] - Biegler, L.T. An overview of simultaneous strategies for dynamic optimization. Chem. Eng. Process. Process Intensif.
**2007**, 46, 1043–1053. [Google Scholar] [CrossRef] - Paul, C.; Bongard, J.C. The road less travelled: Morphology in the optimization of biped robot locomotion. In Proceedings of the 2001 IEEE/RSJ International Conference on Intelligent Robots and Systems, Maui, HI, USA, 29 October–3 November 2001; Volume 1, pp. 226–232. [Google Scholar]
- Matsushita, K.; Yokoi, H.; Arai, T. Pseudo-passive dynamic walkers designed by coupled evolution of the controller and morphology. Robot. Auton. Syst.
**2006**, 54, 674–685. [Google Scholar] [CrossRef] - Ravichandran, T.; Wang, D.; Heppler, G. Simultaneous plant-controller design optimization of a two-link planar manipulator. Mechatronics
**2006**, 16, 233–242. [Google Scholar] [CrossRef] - Portilla-Flores, E.A.; Mezura-Montes, E.; Álvarez-Gallegos, J.; Coello-Coello, C.A.; Cruz-Villar, C.A. Integration of structure and control using an evolutionary approach: An application to the optimal concurrent design of a CVT. Int. J. Numer. Methods Eng.
**2007**, 71, 883–901. [Google Scholar] [CrossRef] - Villarreal-Cervantes, M.G.; Cruz-Villar, C.A.; Alvarez-Gallegos, J.; Portilla-Flores, E.A. Robust structure-control design approach for mechatronic systems. IEEE/ASME Trans. Mechatron.
**2013**, 18, 1592–1601. [Google Scholar] [CrossRef] - Valdez, S.I.; Chávez-Conde, E.; Hernandez, E.E.; Ceccarelli, M. Structure-control design of a mechatronic system with parallelogram mechanism using an estimation of distribution algorithm. Mech. Based Des. Struct. Mach.
**2016**, 44, 58–71. [Google Scholar] [CrossRef] - Villarreal-Cervantes, M.G. Approximate and widespread Pareto solutions in the structure-control design of mechatronic systems. J. Optim. Theory Appl.
**2017**, 173, 628–657. [Google Scholar] [CrossRef] - Villarreal-Cervantes, M.G.; Cruz-Villar, C.A.; Alvarez-Gallegos, J.; Portilla-Flores, E.A. Differential evolution techniques for the structure-control design of a five-bar parallel robot. Eng. Optim.
**2010**, 42, 535–565. [Google Scholar] [CrossRef] - Cervantes-Culebro, H.; Cruz-Villar, C.A.; Peñaloza, M.G.M.; Mezura-Montes, E. Constraint-Handling Techniques for the Concurrent Design of a Five-Bar Parallel Robot. IEEE Access
**2017**, 5, 23010–23021. [Google Scholar] [CrossRef] - Bech, M.M.; Noergaard, C.; Roemer, D.B.; Kukkonen, S. A global multi-objective optimization tool for design of mechatronic components using Generalized Differential Evolution. In Proceedings of the IECON 2016—42nd Annual Conference of the IEEE Industrial Electronics Society, Florence, Italy, 23–26 October 2016; pp. 475–481. [Google Scholar] [CrossRef]
- Storn, R.; Price, K. Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces. J. Glob. Optim.
**1997**, 11, 341–359. [Google Scholar] [CrossRef] - Mezura-Montes, E.; Coello, C.A.C. Constraint-handling in nature-inspired numerical optimization: Past, present and future. Swarm Evol. Comput.
**2011**, 1, 173–194. [Google Scholar] [CrossRef] - Lampinen, J.; Zelinka, I. Mixed integer-discrete-continuous optimization by differential evolution, Part 1: The optimization method. In Proceedings of the 5th International Conference on Soft Computing, Brno, Czech Republic, 9–12 June 1999; pp. 71–76. [Google Scholar]
- Spong, M.W.; Hutchinson, S.; Vidyasagar, M. Robot Modeling and Control; Wiley: New York, NY, USA, 2006; Volume 3. [Google Scholar]
- Goswami, A.; Thuilot, B.; Espiau, B. A study of the passive gait of a compass-like biped robot: Symmetry and chaos. Int. J. Robot. Res.
**1998**, 17, 1282–1301. [Google Scholar] [CrossRef]

**Figure 5.**Structural element parametrization. The label * is assigned for constant geometric values. Additionally, the notation FV = Frontal View and LV = Lateral View is considered.

**Figure 6.**Evolution of the best S-C design execution: (

**a**) Convergence of the mean value of the aggregate function ${J}_{mean}$. (

**b**) Convergence of the mean value of the violated constraints $V{C}_{mean}$.

**Figure 7.**Dynamic behavior of the best solutions of S-C and sequential design processes. Limit cycle of frontal plane dynamics: (

**a**) S-C design. (

**b**) Sequential design. PWS and SPWS limit cycles of the sagittal plane (right leg of SPBR): (

**c**) S-C design. (

**d**) Sequential design.

**Figure 8.**Angular displacements of the left ${\theta}_{left}$ and right ${\theta}_{right}$ leg of the SPBR along with the PWS and SPWS: (

**a**) S-C design. (

**b**) Sequential design. Activation/Deactivation of the semi-passive control signal: (

**c**) S-C design. (

**d**) Sequential design.

**Figure 10.**Behavior of the SPBR mechanical energy along the PWS and SPWS. Potential energy ${U}_{s}$: (

**a**) S-C design. (

**b**) Sequential design. Kinetic energy ${K}_{s}$: (

**c**) S-C design. (

**d**) Sequential design.

Structural Element | Parameter | Value |
---|---|---|

Motor-M | ${m}_{M}$ (kg) | 0.1060 |

${M}_{bl}$ (m) | 0.0747 | |

${M}_{br}$ (m) | 0.0125 | |

${M}_{sl}$ (m) | 0.0125 | |

${M}_{sr}$ (m) | 0.0020 | |

Coupler-C | ${m}_{C}$ (kg) | 0.0037 |

${C}_{ro}$ (m) | 0.0095 | |

${C}_{ri}$ (m) | 0.0020 | |

${C}_{w}$ (m) | 0.0050 | |

Bearing-B | ${m}_{B}$ (kg) | 0.0054 |

${B}_{ro}$ (m) | 0.0065 | |

${B}_{ri}$ (m) | 0.0050 | |

${B}_{w}$ (m) | 0.0126 | |

Shaft-S | ${m}_{S}$ (kg) | 0.0150 |

${S}_{r}$ (m) | 0.0055 | |

${S}_{l}$ (m) | 0.0200 |

Variable | ${\mathit{p}}_{\mathit{max}}$ | ${\mathit{p}}_{\mathit{min}}$ |
---|---|---|

${R}_{f}$ (m) | 0.70 | 0 |

${R}_{s}$ (m) | 0.70 | 0 |

${F}_{l}$ (m) | 0.15 | 0 |

${F}_{w}$ (m) | 0.15 | 0 |

${A}_{l}$ (m) | 0.05 | 0.02 |

${A}_{h}$ (m) | 0.06 | 0.02 |

${A}_{w}$ (m) | 0.08 | 0.04 |

${L}_{l}$ (m) | 0.05 | 0.015 |

${L}_{h}$ (m) | 0.50 | 0 |

${L}_{w}$ (m) | 0.08 | 0.04 |

${H}_{l}$ (m) | 0.2 | 0.02 |

${H}_{r}$ (m) | 0.06 | 0.0075 |

${H}_{cp}$ (m) | 0.30 | 0.04 |

$M{C}_{ro}$ (m) | 0.06 | 0.02 |

Variable | ${\mathit{p}}_{\mathit{wc},\mathit{max}}$ | ${\mathit{p}}_{\mathit{wc},\mathit{min}}$ |
---|---|---|

${\theta}_{fi}$ (rad) | 0.383 | 0.174 |

$\gamma $ (rad) | 0.139 | 0.034 |

**Table 4.**The best results of each algorithm execution. The solution marked in boldface represents the best among runs.

Run | $\mathit{J}\left({\mathit{p}}^{\ast}\right)$ | ${\mathit{J}}_{1}\left({\mathit{p}}^{\ast}\right)$ | ${\mathit{J}}_{2}\left({\mathit{p}}^{\ast}\right)$ | S.D. |
---|---|---|---|---|

1 | 20.1110 | 17.8344 | 2.2766 | 0.1791 |

2 | 19.7594 | 17.2378 | 2.5216 | 0.1915 |

3 | 19.9426 | 14.7218 | 5.2207 | 0.1586 |

4 | 19.6703 | 14.5213 | 5.1489 | 0.2091 |

5 | 22.4318 | 16.4062 | 6.0255 | 0.3124 |

6 | 19.0486 | 16.8770 | 2.1715 | 0.0938 |

7 | 17.8415 | 15.9499 | 1.8915 | 0.2100 |

8 | 22.0788 | 15.2483 | 6.8305 | 0.0260 |

9 | 20.0037 | 17.8671 | 2.1365 | 0.0261 |

10 | 21.0536 | 15.0629 | 5.9906 | 0.0617 |

**Table 5.**Optimal structure and walking condition design variables associated with the best solution obtained by structure-control (S-C) and sequential (Seq.) design processes.

${\mathit{p}}^{\ast}$ | Variable | S-C/Seq. |
---|---|---|

${p}_{sg}^{\ast}$ | ${R}_{f}^{\ast}$ (m) | 0.4266/0.3563 |

${R}_{s}^{\ast}$ (m) | 0.4731/0.5493 | |

${F}_{l}^{\ast}$ (m) | 0.1784/0.1800 | |

${F}_{w}^{\ast}$ (m) | 0.1195/0.1154 | |

${A}_{l}^{\ast}$ (m) | 0.0439/0.0374 | |

${A}_{h}^{\ast}$ (m) | 0.0915/0.0998 | |

${A}_{w}^{\ast}$ (m) | 0.0786/0.0636 | |

${L}_{l}^{\ast}$ (m) | 0.0222/0.0150 | |

${L}_{h}^{\ast}$ (m) | 0.3207/0.3994 | |

${L}_{w}^{\ast}$ (m) | 0.0535/0.0543 | |

${H}_{l}^{\ast}$ (m) | 0.0208/0.0200 | |

${H}_{r}^{\ast}$ (m) | 0.0076/0.0109 | |

${H}_{cp}^{\ast}$ (m) | 0.0690/0.0423 | |

$M{C}_{ro}^{\ast}$ (m) | 0.0221/0.0295 | |

${p}_{sm}^{\ast}$ | ${\rho}_{F}^{\ast}$ (kg/m${}^{3}$) | 450/450 |

${\rho}_{A}^{\ast}$ (kg/m${}^{3}$) | 2700/2700 | |

${\rho}_{L}^{\ast}$ (kg/m${}^{3}$) | 2700/1250 | |

${\rho}_{H}^{\ast}$ (kg/m${}^{3}$) | 2700/2700 | |

${\rho}_{MC}^{\ast}$ (kg/m${}^{3}$) | 450/450 | |

${p}_{wc}^{\ast}$ | ${\theta}_{fi}^{\ast}\left(\right)open="("\; close=")">rad$ | 0.3577/0.1782 |

${\gamma}^{\ast}\left(\right)open="("\; close=")">rad$ | 0.0262/0.0262 |

**Table 6.**Optimal semi-passive control design variables associated with the best solutions obtained by structure-control (S-C) and sequential (Seq.) design processes.

$\mathbf{\Delta}{\mathit{t}}_{{\mathit{k}}_{\mathit{s}}}^{\mathit{u}}\left(\mathbf{s}\right)$ | ${\mathit{\tau}}_{{\mathit{k}}_{\mathit{s}}}\left(\mathbf{Nm}\right)$ | |
---|---|---|

${\mathit{k}}_{\mathit{s}}={\mathit{k}}_{\mathit{s}}^{\mathit{p}}+\mathit{i}$ | S-C/Seq. | S-C/Seq. |

$i=1$ | 0.0890/0.4914 | 0.8271/0.3777 |

$i=2$ | 0.0034/0.3451 | 0.6365/0.1699 |

$i=3$ | 0.0177/0.4621 | 0.9904/0.2288 |

$i=4$ | 0.1039/0.5746 | 0.4626/0.3034 |

$i=5$ | 0.0080/0.5238 | 0.2880/0.3243 |

$i=6$ | 0.0112/0.5610 | 0.4088/0.3798 |

$i=7$ | 0.0033/0.5741 | 0.4450/0.3896 |

$i=8$ | 0.0181/0.5978 | 0.3761/0.3478 |

$i=9$ | 0.0183/0.5727 | 0.1880/0.4162 |

$i=10$ | 0.0061/0.5967 | 0.3825/0.3618 |

$i=11$ | 0.0094/0.5839 | 0.0397/0.5712 |

**Table 7.**Dynamic model parameters of the best solutions obtained by structure-control (S-C) and sequential (Seq.) design processes.

Sagittal plane parameters | ||||||

${m}_{sag}$ (kg) | ${I}_{sag}\phantom{\rule{0.166667em}{0ex}}\left(\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}\right)$ | $b\phantom{\rule{0.166667em}{0ex}}\left(\mathrm{m}\right)$ | $d\phantom{\rule{0.166667em}{0ex}}\left(\mathrm{m}\right)$ | $C{M}_{sag}^{y}\phantom{\rule{0.166667em}{0ex}}\left(\mathrm{m}\right)$ | ${T}_{sag}^{p}\phantom{\rule{0.166667em}{0ex}}\left(\mathrm{s}\right)$ | ${T}_{sag}^{sp}\phantom{\rule{0.166667em}{0ex}}\left(\mathrm{s}\right)$ |

S-C/Seq. | S-C/Seq. | S-C/Seq. | S-C/Seq. | S-C/Seq. | S-C/Seq. | S-C/Seq. |

2.7417/2.0431 | 0.0573/0.0667 | 0.2957/0.3721 | 0.0753/0.0314 | 0.1774/0.1773 | ≈1.2/≈1.2 | ≈1.2/≈1.2 |

Frontal plane parameters | ||||||

${m}_{fro}$ (kg) | ${I}_{fro}\phantom{\rule{0.166667em}{0ex}}\left(\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}\right)$ | $a\phantom{\rule{0.166667em}{0ex}}\left(\mathrm{m}\right)$ | $\varphi \phantom{\rule{0.166667em}{0ex}}\left(\mathrm{rad}\right)$ | $C{M}_{fro}^{y}$ | ${T}_{fro}\phantom{\rule{0.166667em}{0ex}}\left(\mathrm{s}\right)$ | |

S-C/Seq. | S-C/Seq. | S-C/Seq. | S-C/Seq. | S-C/Seq. | S-C/Seq. | |

5.4835/4.0862 | 0.1778/0.1762 | 0.2492/0.1790 | 0.0643/0.0766 | 0.1774/0.1773 | ≈1.2/≈1.2 |

**Table 8.**Numerical results of the aggregate function $J\left({p}^{\ast}\right)$, the walking capability ${J}_{1}\left({p}^{\ast}\right)$ and energetic efficiency ${J}_{2}\left({p}^{\ast}\right)$ design objectives related to the best solutions of structure-control (S-C) and sequential (Seq.) design approaches.

$\mathit{J}\left({\mathit{p}}^{\ast}\right)$ | ${\mathit{J}}_{1}\left({\mathit{p}}^{\ast}\right)$ | ${\mathit{J}}_{2}\left({\mathit{p}}^{\ast}\right)$ |
---|---|---|

S-C/Seq. | S-C / Seq. | S-C / Seq. |

17.8415/48.9470 | 15.9499/7.7350 | 1.8915 / 41.2120 |

**Table 9.**Angular displacement indicators related to the best solutions of structure-control (S-C) and sequential (Seq.) design approaches.

${\mathit{\beta}}_{\mathit{fro}}$ (rad) | $\mathit{max}\left(\right)open="("\; close=")">\left(\right)open="|"\; close="|">{\mathit{\theta}}_{\mathit{f}}\left(\mathit{t}\right)$ (rad) | ${\mathit{\delta}}_{\mathit{sag}}$ (rad) | $\mathit{max}\left(\right)open="("\; close=")">\left(\right)open="|"\; close="|">{\mathit{\theta}}_{\mathit{st}}\left(\mathit{t}\right),\left(\right)open="|"\; close="|">{\mathit{\theta}}_{\mathit{sw}}\left(\mathit{t}\right)$ (rad) |
---|---|---|---|

S-C/Seq. | S-C/Seq. | S-C/Seq. | S-C/Seq. |

0.4392/0.5442 | 0.3577/0.1783 | 0.3031/0.2245 | 0.3003/0.1992 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Martínez-Castelán, J.N.; Villarreal-Cervantes, M.G.
Integrated Structure-Control Design of a Bipedal Robot Based on Passive Dynamic Walking. *Mathematics* **2021**, *9*, 1482.
https://doi.org/10.3390/math9131482

**AMA Style**

Martínez-Castelán JN, Villarreal-Cervantes MG.
Integrated Structure-Control Design of a Bipedal Robot Based on Passive Dynamic Walking. *Mathematics*. 2021; 9(13):1482.
https://doi.org/10.3390/math9131482

**Chicago/Turabian Style**

Martínez-Castelán, Josué Nathán, and Miguel Gabriel Villarreal-Cervantes.
2021. "Integrated Structure-Control Design of a Bipedal Robot Based on Passive Dynamic Walking" *Mathematics* 9, no. 13: 1482.
https://doi.org/10.3390/math9131482