Whole-Body Anti-Input Saturation Control of a Bipedal Robot

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 zero-moment point (ZMP) or center of pressure (CoP) being crucial for balancing [1-4]. The humanoid research community is increasingly interested in the complex motion control tasks of multi-DOF humanoid robots, particularly those with hyper-DOF systems. New developments in locomotion control, tested on quadrupeds and bipeds, often rely on lower-dimensional models or quasi-static assumptions, restricting the robot's dynamic movement [5, 6]. Optimization-based techniques, like non-linear model predictive control (MPC), can be used to plan and control movement while accounting for the robot's entire dynamics. However, if not warm-started, the solver may become trapped in local minima [7, 8].

The center of mass (CoM) of a multi-DOF 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 whole-body control (WBC) is used for dynamic movements. Full-body 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 self-collision 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 real-time optimization by describing robot dynamics as linear constraints with a convex cost function [10-14]. Hirai et al. [15] suggested and successfully implemented ZMP modulation control of CoM on their real humanoid robots. Position-controllable humanoids are proposed in the studies [16, 17], where the concept is extended to whole-body motion control. Hyon et al. [18] suggested the use of torque-controllable robots and a resolved acceleration controller for direct acceleration control of CoM in multi-DOF 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 force-controllable humanoids is proposed in the study [19]. Caron et al. [20] developed a whole-body admittance controller that integrates end-effector 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 anti-input saturation) with low-level control for a bipedal robot that has fewer computations and is preferred for real-time 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 anti-windup double hyperbolic sliding mode controller for a three-legged robot under torque constraints. Researchers are working on developing controllers for actuators with practical limits like saturation, dead zone, and hysteresis to improve closed-loop 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 closed-loop system performance [24-26]. The contributions of our paper are outlined as follows:

• It presents an integration of an anti-input saturation compensator with adaptive control for complex bipedal robot dynamics, focusing on stabilizing ZMP-based bipedal locomotion by limiting torque values at ankle joints.
• It also designs a decentralized adaptive control for high-DOF bipedal robots, as increasing DOFs complicates the inherent nonlinear dynamics and makes control tasks problematic.

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 anti-input 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 [27-30] 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 6-link bipedal robot in the single-support phase of SSP, considering adaptive and non-adaptive 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.

The WBC aims to maintain the balance of a bipedal robot during environmental interaction. It divides motion tasks into trunk (CoM) tasks and lower-limb tasks. The trunk task uses Cartesian-based dynamics and control to stabilize trunk orientation and CoM location. The lower-limb 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 skew-symmetric 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 closed-loop 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 closed-loop 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 positive-definite 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 closed-loop 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 non-negative 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

$\begin{aligned} & \dot{V}_i=\frac{1}{2}\left(\dot{d}_i-2 c_i\right) s_i^2-\mu_i s_i^2-\gamma_i s_i \operatorname{sgn}\left(s_i\right)+  \varepsilon_i s_i-\tilde{w}_{d_i}^T\left(\Omega_{d_i}^{-1} \dot{\hat{w}}_{m_i}+\psi_{d_i} \ddot{q}_{r_i} s_i\right)  -\tilde{w}_{c_i}^T\left(\psi_{c_i} \dot{q}_{v_i} s_i+\Omega_{c_i}^{-1} \dot{\hat{w}}_{c_i}\right)-\tilde{w}_{\kappa_i}^T\left(\varphi_{\kappa_i} s_i+\Omega_{\kappa_i}^{-1} \dot{\hat{w}}_{\kappa_i}\right)\end{aligned}$            (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.

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 whole-body dynamics, integrating ZMP-CoM dynamics with the bipedal multibody system. Further work is needed to address the multi-phase 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 heel-strike and toe-off movements as primary phases. This ability enables biped robots to reproduce human gait patterns, enhancing their walking efficiency and safety. Toe-off and heel-strike actions can modify the system's degree of freedom (DOF), leading to either under- or over-actuation. Consequently, a complete gait cycle is characterized by three phases: full actuation, under-actuation, and over-actuation when comparing DOF values with actuated joints. The challenge with multi-phase locomotion is the discontinuous dynamic response during walking phases due to variations in configurations and dynamical behaviors for each phase. Implementing whole-body dynamics with a floating-base dynamic model represented by the trunk proves to be a robust approach for multi-contact modeling. Thus, upcoming research will concentrate on whole-body motion planning and control while considering the entire gait cycle. It is essential to conduct a comparative analysis between floating-base dynamics and conventional methods that employ a discrete model for each walking sub-phase.