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Edited by: Alexandre Bernardino, Universidade de Lisboa, Portugal

Reviewed by: Kenji Hashimoto, Waseda University, Japan; Kensuke Harada, National Institute of Advanced Industrial Science and Technology, Japan

Specialty section: This article was submitted to Humanoid Robotics, a section of the journal Frontiers in Robotics and AI

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Human beings are highly efficient in maintaining standing balance under the influence of different perturbations. However, biped humanoid robots are far from exhibiting similar skills. This is mainly due to the limitations in the current control and modeling techniques used in humanoid robots. Even though approaches using the Linear Inverted Pendulum Model and the Preview Control schemes have shown improved results, they still suffer from shortcomings in the overall generated motion. We propose here a model and control approach that aims to overcome the limiting assumptions in the LIPM models through using the ankle joint variables in modeling and control of the standing balance of the humanoid robot.

Modeling and control of the biped robot balance and locomotion is a difficult and complex process due to its high dimensionality and non-linearity. Simplifying assumptions are needed in order to facilitate real-time robot control. There have been several studies in modeling biped locomotion. All of these use the zero moment point (ZMP) concept to study and control the motion of the biped. The most notable one is the introduction of the Linear Inverted Pendulum Model by Kajita and Tanie (_{c}_{x}

Fujimoto and Kawamura (

Sugihara and others developed a real-time motion generation method through the control of the whole-body CoM by manipulating the ZMP (Sugihara et al.,

Standing balance recovery after a perturbation occurs is a complex process compromising the use of different balancing strategies. Experimental studies conducted by Horak and Nashner (

Most control approaches for biped robot standing balance recovery implement the ankle strategy to regulate the CoP with fixed feet positions, such as the case in Hirai et al. (

Our main goal here is to propose a new modeling approach based on the Spherical Inverted Pendulum (SIP) Model along with an efficient energy-based control law. In the following section, we will introduce the SIP model for the biped robot. In Section

In the biomechanical literature (Winter,

The ankle joint in the SIP model is assumed to be a two degrees of freedom rotational joint. The CoM is allowed to move in three-dimensional space due to the two rotational degrees of freedom at the ankle. As a result, the CoM can fall anywhere on the surface defined by the hemisphere of radius equal to the stance leg length, as can be seen in Figure

The CoM location in the local frame of reference is defined by ^{T}

As can be seen from Eqs

In order to find a storage function for the SIP model as defined by Willems (

The above system behaves like a spring-mass system with negative stiffness of _{SIP}

The _{SIP}_{SIP}

The solution given by Eq.

However, when the system is disturbed, the above stability condition does not hold and _{p}

The control input,

Using the control torque as the driving input to the linearized system described by Eq.

The approximation

This controller offers the benefits of ease of tuning in comparison to standard proportional derivative controllers, since there is only one gain variable instead of two tunable gains. Furthermore, we will show in the next section that this controller results in a critically damp response, achieving the fastest stabilization time with the least amount of energy consumption possible.

To verify the validity of the energy controller described above, we have run a number of simulation experiments, using classical control approaches, such as PD + Gravity compensation, and the Energy controller presented here. The control torque was constrained to ensure that the CoP does not leave the support polygon defined by the foot. Assuming that the foot is fixed to the floor and the ankle torque does not result in slippage or foot rotation, Yu et al. (

_{X}_{Z}_{Z}

Furthermore, _{Z}

CoM height^{a} |
0.367 |

CoM to hip joint offset^{b} |
0.095 |

Upper leg | 0.11 |

Lower leg | 0.103 |

Foot height | 0.045 |

Foot length | 0.2 |

Total body^{c} |
5 |

Upper leg | 0.591 |

Lower leg | 0.292 |

Foot | 0.296 |

^{a}Pendulum length

^{b}Distance from the CoM as determined by the total robot mass to the hip joint location

^{c}This includes the mass of head and arms

The resulting controlling torque from both control approaches is shown in Figure

From the static kinematic analysis of the inverted pendulum, Flanagan (

Solving Eq. _{CoP}

Both controllers manage to stabilize the SIP in face of impulse disturbances with a maximum sway angle of around 8°, which is well within the limits described in Eq.

In the case of large disturbances, a different strategy needs to be used to avoid falling over. One such strategy is to allow the robot to take a step in order to enlarge its support polygon and prevent a fall. The optimal stepping location is defined using the concept of the capture point (Pratt et al.,

To compute the required orientation of the robot in terms of its ankle angular velocity

For small angular velocities, the biped angle before impact is a linear function of the velocity. A linear approximation can be used to simplify the computation time when implemented on the computer for simulation or on the actual robot hardware. If the angular velocity is larger than ±1 rad/s, the system is forced to take a number of steps before it can reach a complete stop in a balanced state due to size limitation of the biped robot. The exact step length _{s}

Reducing the required energy to maintain the standing balance will prolong the battery life and autonomous behavior of the humanoid robot. Figure

The transfer function from the measurement noise to the output of the closed-loop energy control law is denoted by

The energy control law transfer function, _{Energy}_{P}_{PD}

For the energy control law, the steady-state gain is defined to be:

The classical control law has a similar expression for the steady-state gain as that given by Eq.

For large proportional gains, the steady-state gain of both transfer functions approaches one. In most practical situations, this ideal case cannot be achieved, since the value of these gains is limited to an upper bound. The maximum gain value is imposed by the actuator saturation limits, or in the case of humanoid robots, by the center of pressure constraints. In this situation, the DC gain will be larger than 1, as evident from the denominator of both Eqs

The magnitude of the transfer function for both systems decays to zero as the frequency increases as shown in Figure

To simulate walking at low speeds, the ankle joint of the pendulum is demanded to go back and forth at a rate of two steps per second. A sinusoidal input signal,

The top graph in Figure

To simulate the effects of the real-world measurement and actuation noise, white Gaussian noise was added to the input signal before being fed to the system. The noise signal was filtered at 100 Hz to model the bandwidth of our robot hardware joint position and IMU sensors. This is also the sampling frequency at which the control loop will be running on the robot. The bottom graph of Figure

On the other hand, the energy-based controller tracks the input signal,

The orbital energy of the SIP model defined by Eq. _{SIP}

_{SIP}

_{SIP}

_{SIP}

The solution to the stable condition is given by Eq. _{θ}_{θ}

The maximum acceleration,

By analogy, the saddle point of the system when _{SIP}

The above equation describes the stability bounds on the SIP while using the ankle balancing strategy. The stable region in terms of the SIP states is then defined as

Figure

The SIP model is used to generate and control a stable walking gait using the principles of passive dynamic walking. The SIP model is extended to be described as a hybrid dynamic system with two stages. The first is a continuous dynamic model for the Single Support Phase to describe the CoM behavior. The second stage is during the exchange of support instance and impact of the swing leg. After each impact, an internal impulse push is used to restore any lost energy due to the impact with the ground. In the proposed control architecture, the SIP model and its energy-based control are considered as an internal model that provides the CoM and feet trajectories during the walking cycle. These trajectories are converted to joint motion through a full-body inverse kinematics algorithm to be executed through the robot individual joints. The full control framework is visualized in Figure

The walk cycle stability is a nominally periodic sequence of steps that is stable as a whole, but not locally stable at every instant in time. The motion stability in this way is defined as “the ability to interrupt and avoid a fall.”

In order to generate a walking gait, the authors extended the stepping strategy for standing balance through the injection of a virtual impulse push into the controller at the moment of impact to restore any energy loss. The energy loss is modeled through the angular velocity loss of the pendulum as described below:
^{−} is the pendulum angle just before impact and ^{+} is the angle just after impact and

The motion in the frontal place of the robot is controlled in such a way that will translate the CoM from side to side during the walk. This motion trajectory is dependent on the CoM trajectory in the sagittal plane. This trajectory is provided as a reference input to the energy controller for the CoM motion to follow. Figure

The resulting CoM trajectory is shown in Figure

The vertical motion has the same periodicity as that of the human walk. However, the SIP-generated trajectory has an instantaneous change in the direction of the CoM motion at the moment of foot impact at ∼50% of the gait cycle. This instantaneous change results from the stepping model used with the SIP biped model. The CoM reaches its highest point twice during the gait cycle at around 25 and 75% of the cycle. This is the time when the support leg is fully extended and is in an upright position. The lowest CoM position occurs at the moment of switching support. Figure

This article presents the Spherical Inverted Pendulum model for the biped standing balance control. It describes the system using the two rotational degrees of freedom at the ankle and does not suffer from the limiting assumptions of the traditional approaches such as maintaining a constant CoM height off the ground.

We also described an energy-based control architecture for push recovery on humanoid robots. This control method transforms the SIP model into a critically damped system with the use of a single tuning parameter, so that balance is restored in the fastest and most energy efficient way possible. Finally, a description of the stability bounds and walking gait generation method using the SIP and the energy controller is presented.

AE developed the SIP model and the passivity-based control law and performed the simulations under AP’s direct supervision throughout the research process. AE wrote the manuscript. AP provided editing comments. AP was AE’s first supervisor during his studies.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

This research has been partly funded by the Libyan Higher Education Ministry.