© 2024 The authors. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).
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Bipedal locomotion requires a multilevel control strategy for balance and tracking, with the zeromoment point (ZMP) serving as a heuristic balance criterion. Maintaining the ZMP location within the stability margin indicates stability, but ankle joint actuation behavior restrictions are required. This paper focuses on wholebody control of a bipedal robot, considering control input limitations. Two dynamical models of bipedal dynamics are introduced, integrating centerofmass (CoM)based dynamics with joint space dynamics. The controlled outputs are the CoM position and joint displacements, while the control inputs are the ZMP position and joint torques. Antiinput saturation control is considered to ensure safe values for the control inputs, especially for the ZMP and ankle joint torque signals. A decentralized adaptive approximation control (DAAC) with a saturation compensator is designed. The stability of the proposed controller is proven using Lyapunov theory. Simulation experiments are conducted on a planar 6degreesoffreedom (DOF) bipedal robot. The results demonstrate the robustness of the control architecture even under a disturbance torque of 10 N.m., ensuring safe stability margins for the ZMP and precise tracking for the multiDOF bipedal system.
bipedal locomotion, wholebody control, zeromoment point, decentralized adaptive control, bipedal dynamics
Dynamic mobility of legged robots across difficult terrains requires considering the robot's dynamics, actuation restrictions, and interaction with the environment. Legged robots now focus on movement and balancing, with the zeromoment point (ZMP) or center of pressure (CoP) being crucial for balancing [14]. The humanoid research community is increasingly interested in the complex motion control tasks of multiDOF humanoid robots, particularly those with hyperDOF systems. New developments in locomotion control, tested on quadrupeds and bipeds, often rely on lowerdimensional models or quasistatic assumptions, restricting the robot's dynamic movement [5, 6]. Optimizationbased techniques, like nonlinear model predictive control (MPC), can be used to plan and control movement while accounting for the robot's entire dynamics. However, if not warmstarted, the solver may become trapped in local minima [7, 8].
The center of mass (CoM) of a multiDOF system is a promising controllable variable in Cartesian space for mobile robots, including biped humanoid robots, requiring a direct target location. To address this, a strategy based on wholebody control (WBC) is used for dynamic movements. Fullbody motion control is based on assigning proper degrees of freedom for motion tasks, originating from managing redundant manipulators. It addresses tracking performance, joint limitations, singular configurations, and selfcollision avoidance [9]. WBC allows for decoupling control and motion planning, executing multiple tasks while maintaining the robot's behavior. It uses the robot's entire range of motion and degrees of freedom to distribute motion duties across joints, optimizing tasks while considering actuation, interaction, and system dynamics. WBC uses realtime optimization by describing robot dynamics as linear constraints with a convex cost function [1014]. Hirai et al. [15] suggested and successfully implemented ZMP modulation control of CoM on their real humanoid robots. Positioncontrollable humanoids are proposed in the studies [16, 17], where the concept is extended to wholebody motion control. Hyon et al. [18] suggested the use of torquecontrollable robots and a resolved acceleration controller for direct acceleration control of CoM in multiDOF humanoid robots. By using the weighted pseudoinverse of the Jacobian from ZMP to CoM, the redundancy problem was resolved. A novel task space controller for forcecontrollable humanoids is proposed in the study [19]. Caron et al. [20] developed a wholebody admittance controller that integrates endeffector and CoM methods, along with quadratic programming wrench distribution, to improve walking stabilization in linear inverted pendulum tracking [21]. Ramuzat et al. [22] compared three control techniques on the TALOS Humanoid Robot, focusing on solving locomotion problems such as ascending stairs, walking on flat and uneven ground, and walking on both. The first used a hierarchical quadratic program at the velocity level, the second used a weighted quadratic program called Task Space Inverse Dynamic (TSID) at the acceleration stage, and the last used torque level instead.
In light of the above, the limits of the control inputs are often integrated with the WBC using quadratic programming optimization; however, a few studies have considered bounded control (or antiinput saturation) with lowlevel control for a bipedal robot that has fewer computations and is preferred for realtime applications. Robot control input bounds play a crucial role in maintaining stability and safety during a robot's operation by setting essential limits on the values provided to the actuators. These bounds, which consider factors such as torque, speed, and environmental restrictions, are crucial for regulating the motion of the robot. The maximum and minimum values assigned to each actuator are determined based on physical constraints and performance requirements. Recently, Ghoreishi et al. [23] have proposed an antiwindup double hyperbolic sliding mode controller for a threelegged robot under torque constraints. Researchers are working on developing controllers for actuators with practical limits like saturation, dead zone, and hysteresis to improve closedloop system performance and safety. Two approaches include changing the control effort signal and building an auxiliary system to specify tracking errors, aiming to solve these constraints and enhance closedloop system performance [2426]. The contributions of our paper are outlined as follows:
The study focuses on modeling a bipedal robot using the WBC, integrating centroidal dynamics with joint space dynamics. Control inputs include ground reaction forces (GRFs) and joint torques, while output variables are joint states and CoM states. Bounded control is required due to constraints on the ankle joints and the GRF values. An antiinput saturation control is combined with the proposed control law. A unified control law is designed for joint tracking and compensation of the ZMP error. The control architecture is based on the adaptive function approximation technique, considering uncertainty in bipedal parameters and dynamics. The key idea is that every uncertain term can be represented using orthogonal basis functions such as Chebyshev polynomials, Fourier series, neural network approximators, etc.; see the studies [2730] for more details. The adaptive law for the weighting coefficients is designed using Lyapunov theory to ensure stability based on the passivity condition. Simulation experiments are implemented on a 6link bipedal robot in the singlesupport phase of SSP, considering adaptive and nonadaptive case studies. The results show the effectiveness of the proposed controller.
The rest of the paper is structured as follows: Section 2 introduces WBC dynamic modeling. The bipedal robot's control over locomotion is thoroughly explained in Section 3. Results are presented in Section 4, and conclusions and recommendations for further work are made in Section 5.
This section integrates Cartesian centroid models with joint space dynamics to control bipedal locomotion. Two robot models have been developed: Model 1, utilizing CoM (trunk) dynamics and GRF as control inputs, and Model 2, which reformulates centroid dynamics with ZMP locations as input signals; refer to Figure 1. The point p is the location of the ZMP where the contact force $f_p$ is applied. The base frame is assumed to be placed at the CoM (trunk). The base frame can be positioned at the fixedstance foot with more simplified dynamics. In Model 1, the floating base frame is positioned at the trunk (CoM), while in Model 2, there is a fixed base frame at the stance ankle. The (n+6)DOF dynamics of a bipedal model are decoupled for control tasks aimed at complete control of bipedal locomotion. The following assumptions are imposed [31, 32].
Assumption 1: Due to the brief double support phase, the gait cycle comprises only a single support phase, SSP.
Assumption 2: The ZMP location is maintained within the stance sole by fixing the stance foot. This assumption is crucial to preventing deviations in the ZMP beyond the stability margin.
Assumption 3: There is no variation in the height of the trunk (CoM) during locomotion.
Assumption 4: The bipedal robot is governed by floatingbased dynamics, with the CoM as a floating base with neglected rotational dynamics.
Figure 1. A bipedal model in the single contact locomotion
Remark 1. The main focus of the control problem is to track the desired joint trajectories of a bipedal robot while also keeping within bounds on ZMP values when tracking the desired CoM references. Thus, opting to designate the CoM as a floating base is the optimal approach. By designating the CoM as a floating base and disregarding rotational dynamics, the humanoid system becomes uncoupled, separating the CoM dynamics from the joint space dynamics. Selecting a different component as a floating base can complicate control and motion planning processes. Therefore, if a bipedal system is constructed as a coupled system, it will necessitate additional efforts in motion planning, stabilization, and lowlevel control. Such tasks are not within the scope of this study.
2.1 Model 1
This model is widely used in the literature and includes CoM dynamics combined with link dynamics in joint space with base frame positioned at the trunk or the CoM; see, e.g., [31]. The equation of motion describing the exact fullbody dynamics with a single contact point is expressed as [9, 31, 32].
$\left[\begin{array}{lll}M_{11} & M_{12} & M_{13} \\ M_{21} & M_{22} & M_{23} \\ M_{31} & M_{32} & M_{33}\end{array}\right]\left[\begin{array}{c}\ddot{r}_c \\ \ddot{\omega}_c \\ \ddot{\theta}\end{array}\right]+\left[\begin{array}{c}\eta_1 \\ \eta_2 \\ \eta_3\end{array}\right]=\left[\begin{array}{c}0_3 \\ 0_3 \\ \tau\end{array}\right]+\left[\begin{array}{c}I_3 \\ {\left[r_p^c \times\right]} \\ J^T\end{array}\right] f_p$ (1)
where, $M_{i j}$ is an inertia matrix, $r_c=\left[c_x, c_y, c_z\right]^T$ is the position vector of the floating base, $\omega_c=\left[\omega_x, \omega_y, \omega_z\right]^T$ is the angular velocity of the floating base, $\eta_i$ is an nonlinear vector including the Coriolis, centripetal and gravity effect, $I_3$ is an $3 \times 3$ identity matrix, $0_3$ is a $3 \times 1$ null vector, $\theta \in R^n$ is the angular displacement vector of the target biped, $\tau \in R^{n_a}$ is the control joint torques, $n_a$ is the number of actuated joints, $J \in$ $R^{3 \times n}$ is the Jacobian matrix from the floating base to the contact point $P$.
The following points should be noted:
$\underbrace{\left[\begin{array}{cc}M & 0_{2 \times n} \\ 0_{n \times 2} & H\end{array}\right]}_D \underbrace{\left[\begin{array}{c}\ddot{r}_c \\ \ddot{\theta}\end{array}\right]}_{\ddot{q}}+\left[\begin{array}{l}\eta_1 \\ \eta_2\end{array}\right]=\left[\begin{array}{c}0_{2 \times 1} \\ \tau\end{array}\right]+\left[\begin{array}{c}I_{2 \times 2} \\ 0_{n \times 2}\end{array}\right] f_p=\underbrace{\left[\begin{array}{c}f_p \\ \tau\end{array}\right]}_u$ (2a)
with
$\underbrace{\left[\begin{array}{c}\eta_1 \\ \eta_2\end{array}\right]=\left[\begin{array}{cc}0_{2 \times 2} & 0_{2 \times n} \\ 0_{n \times 2} & C_\theta\end{array}\right]}_C \underbrace{\left[\begin{array}{l}\dot{r}_c \\ \dot{\theta}\end{array}\right]}_{\dot{q}}+\underbrace{\left[\begin{array}{l}g_c \\ g_\theta\end{array}\right]}_g$ (2b)
The following benefits arise from the decoupling of the CoM dynamics from the joint space robot dynamics in Eq. (2):
2.2 Model 2
This model focuses on the bipedal robot as a rooted system at the neglected dynamics stance foot, similar to setting the floating base frame at the grounded stance foot. It includes ZMPCoM dynamics and the robot's joint space dynamics, expressing the following mathematical relationship between ZMP position and CoM [33, 34].
$x_{Z M P}=\left(\frac{c_z}{\ddot{c}_z+g}\right) \ddot{c}_x+c_x\frac{\tau_{c y}}{m_c\left(\ddot{c}_z+g\right)}$ (3a)
$y_{Z M P}=\left(\frac{c_z}{\ddot{c}_z+g}\right) \ddot{c}_y+c_y+\frac{\tau_{c x}}{m_c\left(\ddot{c}_z+g\right)}$ (3b)
where, $g=9.81 \mathrm{~m} / \mathrm{s}^2, p=\left[\begin{array}{lll}x_{Z M P} & y_{Z M P} & z_{Z M P}\end{array}\right]^T$ is the ZMP position vector with $z_{Z M P}$ setting to zero assuming flat ground, and $\left[\begin{array}{lll}\tau_{c x} & \tau_{c y} & \tau_{c z}\end{array}\right]^T$ is the rate of angular momentum about the CoM. Eq. (3) is important in stabilization control of bipedal locomotion and it is similar to the first two equations in Eq. (2) related to the CoM dynamics. Eq. (3) can be reformulated in a matrix form as
$\bar{M} \ddot{c}+g_\alpha=u_c$ (4)
where,
$\begin{gathered}\bar{M}=\left[\begin{array}{cc}\left(\frac{c_z}{\ddot{c}_z+g}\right) & 0 \\ 0 & \left(\frac{c_z}{\tilde{c}_z+g}\right)\end{array}\right], g_\alpha=\left[\begin{array}{c}c_x\frac{\tau_{c y}}{m_c\left(\ddot{c}_z+g\right)} \\ c_y+\frac{\left.\tau_{c x}\right)}{m_c\left(\ddot{c}_z+g\right)}\end{array}\right], u_c=\left[\begin{array}{l}x_{Z M P} \\ y_{Z M P}\end{array}\right]\end{gathered}$
Incorporating Eq. (4) with the bipedal model in joint space dynamics, considering single point contact, i.e., the SSP, yields
$\underbrace{\left[\begin{array}{cc}\bar{M} & 0_{2 \times n} \\ 0_{n \times 2} & \bar{H}\end{array}\right]}_D \underbrace{\left[\begin{array}{c}\ddot{c} \\ \ddot{\theta}\end{array}\right]}_{{\ddot{q}}}+\underbrace{\left[\begin{array}{cc}0_{2 \times 2} & 0_{2 \times n} \\ 0_{n \times 2} & \bar{C}_\theta\end{array}\right]}_C \underbrace{\left[\begin{array}{c}\dot{c} \\ \dot{\theta}\end{array}\right]}_{\dot{q}}+\underbrace{\left[\begin{array}{l}g_\alpha \\ \bar{g}_\theta\end{array}\right]}_g=\left[\begin{array}{c}0_2 \\ \tau\end{array}\right]+\left[\begin{array}{c}I_{2 \times 2} \\ 0_{n \times 2}\end{array}\right] u_c=\underbrace{\left[\begin{array}{c}u_c \\ \tau\end{array}\right]}_u$ (5)
The following points should be noted, despite the similarities between Eqs. (2) and (5):
The WBC aims to maintain the balance of a bipedal robot during environmental interaction. It divides motion tasks into trunk (CoM) tasks and lowerlimb tasks. The trunk task uses Cartesianbased dynamics and control to stabilize trunk orientation and CoM location. The lowerlimb task controls the trajectory of the swing foot using joint space dynamics to position the foot with sufficient clearance from the ground. In the following, a decentralized adaptive control based on passivity is described in detail to stabilize and track the bipedal motion. Recalling Eqs. (2) and (5), which can be expressed in a unified form as
$D \ddot{q}+C \dot{q}+g=u$ (6)
with nomenclature defined in (2) and (5). The system in (6) has the following properties and assumptions.
Property 1. Uniform bounds apply to the gravity vector, the Coriolis, centrifugal, and inertia matrices.
These bounds on physical parameters can be determined using a CAD/CAM model for the bipedal robot or using one of the system identification approaches; see the study [33] for more details.
Property 2. If $C(q, \dot{q})$ is specified using the Christoffel symbols then the matrix $N=\dot{D}2 C$ is a skewsymmetric matrix.
Control input signals in Eq. (2) are subjected to the following constraints:
$u_i=\operatorname{sat}\left(\beta_i\right)=\left\{\begin{array}{c}u_{i_{\max }}, \beta_i \geq u_{i_{\max }} \\ \beta_i, u_{i_{\min }}<\beta_i<u_{i_{\max }} \\ u_{i_{\min }}, \beta_i \leq u_{i_{\min }}\end{array}, i=1,2,3, \ldots, m\right.$ (7)
The control tolerance vector's components, $\alpha_i$, can be expressed as
$\alpha_i=\left\{\begin{array}{c}u_{i_{\max }}\beta_i, \beta_i \geq u_{i_{\max }} \\ 0, u_{i_{\min }}<\beta_i<u_{i_{\max }} \\ u_{i_{\min }}\beta_i, \beta_i \leq u_{i_{\min }}\end{array}\right.$ (8)
A decoupled formulation of Eq. (6) is as follows:
$d_{i i}(q) \ddot{q}_i+c_{i i}(q, \dot{q}) \dot{q}_i+\kappa_i(q, \dot{q})=u_i$ (9a)
with
$\kappa_i(q, \dot{q})=\sum_{\substack{j=1 \\ j \neq i}}^m d_{i j}(q) \ddot{q}_j+\sum_{\substack{j=1 \\ j \neq i}}^m c_{i j}(q, \dot{q}) \dot{q}_j+g_i(q)$ (9b)
For uncertain dynamics, the following control law is chosen:
$\begin{aligned} & \hat{d}_{i i}(q) \ddot{q}_{r_i}+\hat{c}_{i i}(q, \dot{q}) \dot{q}_{r_i}+\mu_i S_i+\hat{\kappa}_i(q, \dot{q}) +\gamma_i \operatorname{sgn}\left(s_i\right)=u_{i_i}=\hat{\alpha}_i+\beta_i\end{aligned}$ (10a)
where,
$\dot{q}_{r_i}=\dot{q}_{d_i}+\lambda_i\left(q_{d_i}q_i\right)$ (10b)
where, $\mu_i$ is a positive feedback gain, $\lambda_i$ is a time constant parameter, $\gamma_i$ is a sliding term gain and the subscripts $\mathrm{r}$ and $\mathrm{d}$ stands for the required and desired references. To estimate the uncertainty in Eq. (10), adaptive control based on FAT is utilized. The uncertainty is estimated using an orthogonal polynomial approximator, presuming that the uncertain dynamic matrices and vectors are functions of time. Thus, the control law in Eq. (11) becomes
$\begin{aligned} & \underbrace{\hat{w}_{d_i}^T \psi_{d_i}}_{d_i} \ddot{q}_{r_i}+\underbrace{\hat{w}_{c_i}^T \psi_{c_i}}_{c_i} \dot{q}_{r_i}+\eta_i s_i+\underbrace{\hat{w}_{\kappa_i}^T \psi_{\kappa_i}}_{\kappa_i}\underbrace{\hat{w}_{\alpha_i}^T \psi_{\alpha_i}}_{\alpha_i} +\gamma_i \operatorname{sgn}\left(s_i\right)=\beta_{d_i}\end{aligned}$ (11)
where, $w_{(.)} \in R^{n_o}$ and $\psi_{(.)} \in R^{n_o}$ represents the weighting coefficients and orthogonal basis function vectors, respectively, with $n_o$ denoting the number of basis function terms. The following closedloop dynamics result from deducting Eq. (11) from Eq. (9):
$\begin{aligned} & d_i \dot{s}_i+c_i s_i+\mu_i s_i+\gamma_i \operatorname{sgn}\left(s_i\right)= \widetilde{w}_{d_i}^T \psi_{d_i} \ddot{q}_{r_i}\widetilde{w}_{c_i}^T \psi_{c_i} \dot{q}_r\widetilde{w}_{\kappa_i}^T \psi_{\kappa_i}+\widetilde{w}_{\alpha_i}^T \psi_{\alpha_i} +\left(\beta_i\beta_{d_i}\right)+\varepsilon_i\end{aligned}$ (12)
where, $\varepsilon_i$ is error in the approximation. Achieving stable closedloop dynamics requires the selection of the following updated adaptive laws.
$\begin{aligned} & \dot{\hat{w}}_{d_i}=\Omega_{d_i} \psi_{d_i} \ddot{q}_{r_i} s_i \\ & \dot{\hat{w}}_{c_i}=\Omega_{c_i} \psi_{c_i} \dot{q}_{r_i} s_i \\ & \dot{\hat{w}}_{\kappa_i}=\Omega_{\kappa_i} \psi_{\kappa_i} s_i \\ & \dot{\hat{w}}_{\alpha_i}=\Omega_{\alpha_i} \psi_{\alpha_i} s_i\end{aligned}$ (13)
where, $\Omega_{(.)} \in R^{n_o \times n_o}$ is the adaptation's positivedefinite gain matrix. The stability of the suggested control structure is demonstrated by the following theorem.
Theorem 1. Given $\beta_{d_i}=\beta_i$, the dynamics of the bipedal subsystems in (9), the control laws in Eq. (10), the corresponding closedloop dynamics in Eq. (12), and the associated update adaptive laws in Eq. (13) are all stable in light of Barbalat's lemma [27].
Proof.
Consider the following nonnegative function, $V_i$, for the ith subsystem:
$\begin{aligned} & V_i=\frac{1}{2} d_i s_i^2+\frac{1}{2} \widetilde{w}_{d_i}^T \Omega_{d_i}^{1} \widetilde{w}_{d_i}+\frac{1}{2} \widetilde{w}_{c_i}^T \Omega_{c_i}^{1} \widetilde{w}_{c_i} +\frac{1}{2} \widetilde{w}_{\kappa_i}^T \Omega_{\kappa_i}^{1} \widetilde{w}_{\kappa_i}+\frac{1}{2} \widetilde{w}_{\alpha_i}^T \Omega_{\alpha_i}^{1} \widetilde{w}_{\alpha_i}\end{aligned}$ (14)
Calculating the time derivative of Eq. (14) and inserting Eq. (12) into it yields
$\begin{aligned} & \dot{V}_i=\frac{1}{2}\left(\dot{d}_i2 c_i\right) s_i^2\mu_i s_i^2\gamma_i s_i \operatorname{sgn}\left(s_i\right)+ \varepsilon_i s_i\tilde{w}_{d_i}^T\left(\Omega_{d_i}^{1} \dot{\hat{w}}_{m_i}+\psi_{d_i} \ddot{q}_{r_i} s_i\right) \tilde{w}_{c_i}^T\left(\psi_{c_i} \dot{q}_{v_i} s_i+\Omega_{c_i}^{1} \dot{\hat{w}}_{c_i}\right)\tilde{w}_{\kappa_i}^T\left(\varphi_{\kappa_i} s_i+\Omega_{\kappa_i}^{1} \dot{\hat{w}}_{\kappa_i}\right)\end{aligned}$ (15)
Combining the adaptive laws in Eq. (13) with the passivity property results in
$\dot{V}_i=\mu_i s_i^2\gamma_i s_i \operatorname{sgn}\left(s_i\right)+\varepsilon_i s_i$ (16)
Assume that $\gamma_i \geq\left\varepsilon_i\right+\sigma_i$, it leads to
$\dot{V}_i \leq\mu_i s_i^2\sigma_i s_i<0$ (17)
Eq. (17) is asymptotically stable according to Barbalat’s lemma.
Remark 2. The control input vector in Eq. (6) consists of ZMP coordinates and joint torques. To maintain stable and balanced locomotion, the ZMP location within the stance foot is crucial; otherwise, the bipedal system may become unstable. Hence, the ZMP bounds are set based on the foot dimension with a safety margin. Ankle joint torque significantly influences the robot's balance during locomotion as it directly correlates with the ZMP location; hence, the torque must be restricted to guarantee safe motion. Refer to pages 98 of the study [32] and the study [35] for further information.
The section conducts some simulation experiments to validate the proposed control architecture on a planar 6link bipedal robot shown in Figure 2, using physical parameters from Table 1.
The simulation experiments utilized the MATLAB/SIMULINK 2023b package with a fixed time step size of 0.01s employing the DormandPrince (RK8) formula offering eighthorder accuracy. There are two options for groundfoot contact. One approach involves establishing a compliant contact between the stance foot and ground using a massspringdamper model, necessitating an assessment of ground stiffness and damping as outlined in the studies [3638]. Alternatively, a simplified model assumes the bipedal robot is affixed to the massless stance foot, disregarding ankle height. In this scenario, the ground interacts with the bipedal robot through the actuated ankle joint, a technique adopted in our simulations. Onelevel control is intended to stabilize bipedal locomotion, which is an intriguing point. The desired gait patterns are 0° for the swing foot and 90° for the legs, thighs, and trunk links. When examining the sagittal plane alone, this position satisfies the equilibrium for bipedal motion. For simulation, two scenarios are chosen: Scenario 1 addresses tracking control and stabilization, taking input saturation into account, and Scenario 2 makes the assumption of an arbitrary disturbance force in the presence of an input saturation compensator. An impulse torque of 10 N.m. is applied to the bipedal model to perturb the dynamic response of the bipedal system.
Table 1. Physical specifications of the bipedal robot
Components 
Moments of Inertia [kgm^{2}] 
Lengths [m] 
Masses [kg] 
Foot 
0.016 
0.3 
2 
Shank 
0.06 
0.45 
3.6 
Thigh 
0.06 
0.45 
3.7 
Trunk 
0.145 
0.45 
10 
The following assumptions are imposed in the simulation:
Figure 2. The CoM position error and the ZMP control inputs considering three simulation scenarios
Figures 3 and 4 show the CoM position error and the ZMP position signals, respectively. The ZMP position is considered here as a control input with boundary limits based on the size of the stance foot. The period of the swing phase is taken as 0.5 seconds. It should be noted that the position error for the CoM will be larger in the presence of a disturbance torque with a larger value for control inputs. However, the availability of saturation compensators solves the problem of ensuring the ZMP limits within the stable region. The disturbance force causes an unrealistic shift in the ZMP location, indicating that the system is beyond the stability margin and unstable. However, the position error of the CoM approaches zero after 0.4 s; see Figure 3. Nonetheless, bounded control is crucial for stabilizing the system. The position error with the saturated control inputs for the biped joints is depicted in Figures 3 and 4 respectively. The position error asymptotically approaches zero in the steadystate region. In fact, the tracking position error is increased by the use of the antiinput saturation compensator. Nonetheless, its presence is crucial in achieving stable locomotion, as the constrained ankle torque values are relevant to ZMP stability. The CADCAM model proves to be beneficial in offering a comprehensive insight into the system's dynamic response and certain physical parameters of the target system. In our simulation tests, the joint torque limitations are arbitrarily chosen to assess the efficiency of our proposed controller, ranging from 150 to 150 N.m. Several key factors influence the control input limits, including foot size, ground incline, and the CoM location of the target bicycle, determined by system configurations and modeling. These factors impact the ZMP limit, ankle torque values, as well as the robot's speed.
Figure 3. Position error for biped joints considering three scenarios
Figure 4. Control inputs considering three scenarios. The limits on the joint actuators are set to 150 N.m. to 150 N.m.
The following important points should be noted:
This paper discusses the development of a decentralized adaptive approximation control that addresses saturation problems in control inputs. The key idea is to design a unified control based on wholebody dynamics, integrating ZMPCoM dynamics with the bipedal multibody system. Further work is needed to address the multiphase gait cycle and solve discontinuous dynamic responses. The integration of a saturation compensator with the control law enhances balance and safe locomotion. The human locomotion system is seen as a valuable model for biped robots due to its similarities. Recent advancements in biped robotics are centered on achieving robotic walking that closely mimics human gait. Human gait consists of two main phases: stance (DSP) and swing (SSP). Foot rotation is a key characteristic of human walking, involving heelstrike and toeoff movements as primary phases. This ability enables biped robots to reproduce human gait patterns, enhancing their walking efficiency and safety. Toeoff and heelstrike actions can modify the system's degree of freedom (DOF), leading to either under or overactuation. Consequently, a complete gait cycle is characterized by three phases: full actuation, underactuation, and overactuation when comparing DOF values with actuated joints. The challenge with multiphase locomotion is the discontinuous dynamic response during walking phases due to variations in configurations and dynamical behaviors for each phase. Implementing wholebody dynamics with a floatingbase dynamic model represented by the trunk proves to be a robust approach for multicontact modeling. Thus, upcoming research will concentrate on wholebody motion planning and control while considering the entire gait cycle. It is essential to conduct a comparative analysis between floatingbase dynamics and conventional methods that employ a discrete model for each walking subphase.
$\left(c_x, c_y, c_z\right)$ 
the coordinates of the floating base 
CoM 
center of mass 
CoP 
center of pressure 
DAAC 
a decentralized adaptive approximation control 
DOF 
degrees of freedom 
DSP 
double support phase 
FAT 
function approximation technique 
GRF 
ground reaction force 
MPC 
model predictive control 
SSP 
single support phase 
WBC 
whole body control 
$c_\theta$ 
a Coriolis and centripetal matrix related to the joint space dynamics 
$f_p$ 
a ground reaction force $\in R^3$ applied to point p 
H 
an inertia submatrix associated with joint space dynamics 
$I_{n \times n}$ 
an identity matrix 
M 
an inertia matrix associated with com dynamics 
$M_{i j}$ 
an inertia submatrix 
n 
number of degrees of freedom 
$n_o$ 
number of terms of an orthogonal basis function 
$n_a$ 
number of actuated joints 
$O_{n \times n}$ 
a null matrix 
p 
position vector of the ZMP coordinates 
$V_i$ 
Lyapunov function 
$r_c$ 
a positive vector $\in R^3$ of a floating base 
$s_i$ 
a sliding velocity variable 
$u_i$ 
control input for subsystem i 
$w_{(.)}$ 
weightcoefficient vector $\in R^{n_o}$ 
ZMP 
zero moment point 
Greek symbols 

$\alpha_i$ 
a control tolerance parameter 
$\beta_i$ 
a control input signal without saturation 
$\varepsilon_i$ 
modeling error 
$\eta_i$ 
a nonlinear term vector including Coriolis, centripetal and gravity effects 
$\lambda_i$ 
a time constant gain 
$\tau_{c i}$ 
a rate of angular momentum component about the COM 
$\psi_i$ 
an orthogonal basis function vector $\in R^{n_0}$ 
$\psi_i$ 
angular velocity vector $\in R^3$ of a floating base 
$\theta$ 
angular displacement vector $\in R^n$ of the biped joints 
$\tau$ 
a joint torque vector $\in R^{n_a}$ 
$\Omega_{(.)}$ 
an adaptation positive gain matrix $\in R^{n_0 \times n_0}$ 
Subscripts 

c 
floating base 
d 
desired reference 
r 
required filtered reference 
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